ZHANG WENPENG, LI JUNZHUANG, LIU DUANSEN (EDITORS) RESEARCH ON SMARANDACHE PROBLEMS IN NUMBER THEORY (VOL. II)

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1 ZHANG WENPENG, LI JUNZHUANG, LIU DUANSEN EDITORS RESEARCH ON SMARANDACHE PROBLEMS IN NUMBER THEORY VOL. II PROCEEDINGS OF THE FIRST NORTHWEST CONFERENCE ON NUMBER THEORY Hexis Phoenix, AZ 005

2 EDITORS: ZHANG WENPENG DEPARTMENT OF MATHEMATICS NORTHWEST UNIVERSITY XI AN, SHAANXI, P.R.CHINA LI JUNZHUANG, LIU DUANSEN INSTITUTE OF MATHEMATICS SHANGLUO TEACHER S COLLEGE SHANGLUO, SHAANXI, P.R.CHINA RESEARCH ON SMARANDACHE PROBLEMS IN NUMBER THEORY VOL. II PROCEEDINGS OF THE FIRST NORTHWEST CONFERENCE ON NUMBER THEORY Hexis Phoenix, AZ 005

3 This book can be ordered in a paper bound reprint from: Books on Demand ProQuest Information & Learning University of Microfilm International 300 N. Zeeb Road P.O. Box 346, Ann Arbor MI , USA Tel.: Customer Service Peer Reviewers: Shigeru Kanemitsu, Graduate School of Advanced Technology, University of Kinki, -6 Kayanomori, Iizuka , Japan. Zhai Wenguang, Department of Mathematics, Shandong Teachers' University, Jinan, Shandong, P. R. China. Guo Jinbao, College of Mathematics and Computer Science, Yanan University, Yanan, Shaanxi, P. R. China. Copyright 005 by Hexis and Zhang Wenpeng, Li Junzhuang, Liu Duansen, and all authors for their own articles Many books can be downloaded from the following E-Library of Science: ISBN: Standard Address Number: Printed in the United States of America

4 Contents Dedication Preface v xi On the Smarandache m-th power residues Zhang Wenpeng An equation involving Euler s function 5 Yi Yuan On the Function ϕn and δ k n 9 Xu Zhefeng On the mean value of the Dirichlet s divisor function in some special sets 3 Xue Xifeng A number theoretic function and its mean value 9 Ren Ganglian On the additive k-th power complements 3 Ding Liping On the Smarandache-Riemann zeta sequence 9 Li Jie On the properties of the hexagon-numbers 33 Liu Yanni On the mean value of the k-th power part residue function 37 Ma Jinping On the integer part of the m-th root and the k-th power free number 4 Li Zhanhu, A Formula For Smarandache LCM Ratio Sequence 45 Wang Ting On the m-th power complements sequence 47 Yang Hai, Fu Ruiqin On the Smarandache ceil function and the Dirichlet divisor function 5 Ren Dongmei

5 viii RESEARCH ON SMARANDACHE PROBLEMS IN NUMBER THEORY II On a dual function of the Smarandache ceil function 55 Lu Yaming On the largest m-th power not exceeding n 59 Liu Duansen and Li Junzhuang On the Smarandache back concatenated odd sequences 63 Li Junzhuang and Wang Nianliang On the additive hexagon numbers complements 7 Li chao and Yang Cundian On the mean value of a new arithmetical function 75 Yang Cundian and Liu Duansen On the mean value of a new arithmetical function 79 Zhao Jian and Li Chao On the second class pseudo-multiples of 5 sequences 83 Gao Nan A class of Dirichlet series and its identities 87 Lou Yuanbing An arithmetical function and the perfect k-th power numbers 9 Yang Qianli and Yang Mingshun An arithmetical function and its mean value formula 95 Ren Zhibin and Zhao Xiaopeng On the square complements function of n! 99 Fu Ruiqin, Yang Hai On the Smarandache function 03 Wang Yongxing Mean value of the k-power complement sequences 07 Feng Zhiyu On the m-th power free number sequences Chen Guohui A number theoretic function and its mean value 5 Du Jianli On the additive k-th power part residue function 9 Zhang Shengsheng An arithmetical function and the k-th power complements 3 Huang Wei On the triangle number part residue of a positive integer 7 Ji Yongqiang

6 Contents ix An arithmetical function and the k-full number sequences 3 Zhao Jiantang On the hybrid mean value of some special sequences 35 Li Yansheng and Gao Li On the mean value of a new arithmetical function 43 Ma Junqing

7 This book is dedicated to Professor Florentin Smarandache, who listed many new and unsolved problems in number theory.

8 Preface Arithmetic is where numbers run across your mind looking for the answer. Arithmetic is like numbers spinning in your head faster and faster until you blow up with the answer. KABOOM!! Then you sit back down and begin the next problem. Alexander Nathanson Mathematics is often referred to as a servant of science and number theory as a queen of mathematics, by which we are to understand that number theory should contribute to the development of science by continuously supplying challenging problems and illustrative examples of its own discipline in anticipation of later practical use in the real world. The research on Smarandache Problems plays a key role in the development of number theory. Therefore, many mathematicians show their interest in the Smarandache problems and they conduct much research on them. Under such circumstances, we published the book <RESEARCH ON SMARANDACHE PROBLEMS IN NUMBER THEORY>, Vol. I, in September, 004. That book stimulated more Chinese mathematicians to pay attention to Smarandache conjectures, open and solved problems in number theory. The First Northwest Number Theory Conference was held in Shangluo Teacher's College, China, in March 005. One of the sessions was dedicated to the Smarandache problems. In that session, several professors gave a talk on Smarandache problems and many participants lectured on Smarandache problems both extensively and intensively. This book includes 34 papers, most of which were written by participants of the above mentioned conference. All these papers are original and have been refereed. The themes of these papers range from the mean value or hybrid mean value of Smarandache type functions, the mean value of some famous number theoretic functions acting on the Smarandache sequences, to the convergence property of some infinite series involving the Smarandache type sequences. We sincerely thank all the authors and the referees for their important contributions. Thanks are also due to Dr. Xu Zhefeng for his effort of making files of LaTeX style. The last, but not the least, thanks are due to the teachers and students of Shangluo Teacher's College in China for their great help in our successful conference. August 0, 005 Zhang Wenpeng, Li Junzhuang, Liu Duansen

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10 ON THE SMARANDACHE M-TH POWER RESIDUES Zhang Wenpeng Department of Mathematics, Northwest University, Xi an, Shaanxi, P.R.China Abstract Keywords: For any positive integer n, let φn be the Euler function, and a mn denotes the Smarandache m-th power residues function of n. The main purpose of this paper is using the elementary method to study the number of the solutions of the equation φn = a mn, and give all solutions for this equation. Arithmetical function; Equation; Solutions.. Introduction For any positive integer n, let n = p α pα pα k k denotes the factorization of n into prime powers. The Smarandache m-th power residues function a m n are defined as a m n = p β pβ pβ k k, β i = min{m, α i }, i =,., k. In problem 65 of [], Professor F.Smarandache asked us to study the properties of this function. Let φn denotes the Euler function. That is, φn denotes the number of all positive integers not exceeding n which are relatively prime to n. It is clear that φn and a m n both are multiplicative functions. In this paper, we shall use the elementary method to study the solutions of the equation involving these two functions, and give all solutions for it. That is, we shall prove the following: Theorem. Let m be a fixed integer with m. Then the equation φn = a m n have m + solutions, namely. Proof of the theorem n =, m, α 3 m, α =,,, m. In this section, we shall complete the proof of the theorem. Let n = p α pα pα k k denotes the factorization of n into prime powers, then from the definitions of φn and a m n we have a m n = p β pβ pβ k k, β i = min{m, α i }, i =,., k This work is supported by N.S.F. of P.R.China 07093,

11 RESEARCH ON SMARANDACHE PROBLEMS IN NUMBER THEORY II and φn = p α p p α p p α k k p k. It is clear that n = is a solution of the equation φn = a m n. If n >, then we will discuss the problem in four cases: i If α i > m, then we have α i m, max {β j} m. Combining j k and, we know that φn a m n. That is, there is no any solution satisfied φn = a m n in this case; ii If α = α = = α k = m, then we have and a m n = p m p m p m k φn = p m p p m p p m k p k. It is clear that only n = m is a solution of the equation in this case; iii If max i k {α i} < m, then from and, and noting that β k = α k > α k, we know that φn a m n for all n in this case; iv If i, j such that α i = m and α j < m, then from, and equation φn = a m n, we get p =. If not, then φn is an even number, but a m n is an odd number. Noting that if α k < m, then φn a m n, so we have α k = m and φn = α p α p p m k p k = a m α a m p α a mp m k = a m n. If α = m, then n = m is the case of ii. Hence, α < m. Now from the equation, we have p α p p m k p k = a m p α a mp m k. 3 Noting that p i if i >, from 3 we can deduce that only one term with the form p i in the left side of 3. That is, From n = α p m. φ α p m = a m α p m p m p = p m, we get n = α 3 m, α < m. That is, n = α 3 m, α < m are all the solutions of φn = a m n in this case. Now combining the above four cases we may immediately get all m + solutions of equation φn = a m n, namely n =, m, α 3 m, α =,., m. This completes the proof of Theorem.

12 On the Smarandache m-th power residues 3 References [] F. Smarandache. Only Problems, Not Solutions. Chicago: Xiquan Publishing House, 993, pp54. [] Tom M. Apostol. Introduction to Analytic Number Theory. New York: Springer-Verlag, 976.

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14 AN EQUATION INVOLVING EULER S FUNCTION Yi Yuan Research Center for Basic Science, Xi an Jiaotong University, Xi an, Shaanxi, P.R.China yuanyi@mail.xjtu.edu.cn Abstract Keywords: In this paper, we use the elementary methods to study the number of the solutions of an equation involving the Euler s function φn, and give an interesting identity. Multiplicative function; Arithmetical property; Identity.. Introduction For any positive integer n, the Euler function φn is defined to be the number of all positive integers not exceeding n which are relatively prime to n. We also define Jn as the number of all primitive Dirichlet s characters mod n. Let A denotes the set of all positive integers n satisfying the equation φ n = njn. In this paper, we using the elementary methods to study the convergent properties of a new Dirichlet s series involing the solutions of the equation φ n = njn, and give an interesting identity for it. That is, we shall prove the following conclusion: Theorem. For any real number s >, we have the identity n s = ζsζ3s, ζ6s n= n A where ζs is the Riemann zeta-function. Taking s = and, and note that ζ = π /6, ζ4 = π 4 /90, ζ6 = π 6 /945, ζ = 69π / , we may immidiately deduce the following identities: n= n A n = 35 π 4 ζ3 and n= n A n = π. This work is supported by N.S.F. of P.R.China 07093

15 6 RESEARCH ON SMARANDACHE PROBLEMS IN NUMBER THEORY II. Proof of the theorem In this section, we will complete the proof of the theorem. First note that both φn and Jn are multiplicative functions, and if n = p α, then Jp = p, φp = p, Jp α = p α p and φp α = p α p for all positive integer α > and prime p. So we will discuss the solutions of the equation φ n = njn into three cases. a It is clear that n = is a solution of the equation φ n = njn. b If n >, let n = p α pα pαs s denotes the prime powers decomposition of n with all α i i =,,, k, then we have Jn = p α p p α k k p k and φ n = p α p p α p p α k k p k. For this case, n is also a solution of the equation φ n = njn. c If n = p p p r p α r+ r+ pα k k with p < p < < p r and α j >, j = r +, r +,, k, then from b and the definition of φn and Jn we have φ n = njn if and only if or φ p p p r = p p p r Jp p p r p p p r = p p p r p p p r. It is clear that p r can not divide p p p r. So the equation φ n = njn has no solution in this case. Combining the above three cases we may immediately obtain the set of all solutions of the equation φ n = njn is n = p α pα and. That is, A is the set of all square-full numbers and. Now we define the arithmetical function an as follows: an = For any real number s > 0, it is clear that {, if n A, 0, if otherwise. n= n A n s < n= n s, pαs s with all α i > and is convergent if s >. So for s, from the Euler product ns n= formula see [],we have

16 An equation involving Euler s function 7 n= n s = p = p = p = p = p = p = p + ap p s + ap3 p 3s + + p s + + p3s + p s p s p s p s + p s p s p 3s + p s p s p 6s p s p s p 3s p 6s p s p 3s = ζsζ3s, ζ6s where ζs is the Riemann zeta-function, and p denotes the product over all primes. This completes the proof of the theorem. References [] F.Smaradache. Only problems, not solutions, Xiquan Publishing House, Chicago, 993. [] Tom M. Apostol. Introduction to Analytic Number Theory. New York: Springer-Verlag, 976.

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18 ON THE FUNCTION ϕn AND δ K N Xu Zhefeng Department of Mathematics, Northwest University, Xi an, Shaanxi, P.R.China zfxu@nwu.edu.cn Abstract Keywords: The main purpose of this paper is using the elementary method to study the divisibility of δ k n by ϕn, and find all the n such that ϕn δ k n. Euler function; Arithmetical function; Divisibility.. Introduction and main results For a fixed positive integer k and any positive integer n, we define a new arithmetic function as following: δ k n = max{d d n, d, k = } If n the Euler function ϕn is defined to be the number of positive integers not exceeding n which are relatively prime to n; thus, where n k= ϕn = n k= indicates that the sum is extended over those k relatively prime to n. In this paper, we shall study the divisibility of δ k n by ϕn, and find all the n such that ϕn δ k n. In fact, we shall prove the following result: Theorem. 0, α, β N. ϕn δ k n if and only if n = α 3 β, where α > 0, β. Proof of the theorem In this section, we will complete the proof of the theorem. First we need the following Lemma. This work is supported by N.S.F. of P.R.China

19 0 RESEARCH ON SMARANDACHE PROBLEMS IN NUMBER THEORY II Lemma For n, we have ϕn = n p n. p Proof. See Theorem.4 in reference []. Now we use the above lemma to complete the proof of Theorem. We will discuss it in two cases. I. n, k =, then δ k n = n. Let n = p α pα pαs s is the factorization of n into prime powers. From the Lemma, we can write If ϕn δ k n, then that is ϕn = p α p p α p p αs s p s, p α p p α p p αs s p s n, p p p s p p p s, from the above formula, we are sure of i p =, if not, p p p s is even but p p p s odd and ii s, since p i i =, 3,, s but p p 3 p s can not be divided by. If s =, then n = α, α. If s =, then n = α 3 β, α, β >. Thus, we can obtain that n = α 3 β, α, β 0 such that ϕn δ k n when n, k =. II. n, k, we can write n = n n, where n, k = and n, n =, then δ k n = n, and ϕn = ϕn ϕn, If ϕn δ k n, that means ϕn ϕn n, that is from, we can get ϕn n ϕn n n = α 3 β, 3

20 On the Function ϕn and δ k n where α, β 0, α, β N. combining 3and4, we can easily get this means Otherwise n, n. So ϕn α 3 β, ϕn =. n = Altogether, whether n, k = or not, we can obtain n = α 3 β, α, β 0 such that ϕn δ k n. This completes the proof of the theorem. References [] Ibstedt. Surfinig On the Ocean of Numbers- A Few Smarandache Notions and Similar Topics, Erhus University Press, New Mexico. [] T.M.Apostol. Introduction to Analytic Number Theory, Springer-verlag, New York, 976. [3] F.Mertens. Ein Beitrag Zur Analytischen Zahlentheorie, Crelle s Journal, 78874, 46-6.

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22 ON THE MEAN VALUE OF THE DIRICHLET S DIVISOR FUNCTION IN SOME SPECIAL SETS Xue Xifeng Department of Mathematics, Northwest University, Xi an, Shaanxi, P.R.China Abstract Keywords: The main purpose of this paper is using the analytic methods to study the mean value properties of the Dirichlet s divisor function in some special sets, and give several interesting asymptotic formulae for them. Divisor function; Mean value; Asymptotic formula.. Introduction For any positive integer n and k, the k-th power complement function b k n of n is the smallest positive integer such that nb k n is a perfect k-th power. In problem 9 of reference [], Professor F.Smarandache asked us to study the properties of this function. About this problem, some people had studied it before, and obtained some interesting conclusions, see references [4] and [5]. In this paper, we define two new sets B = {n N, b k n n} and C = {n N, n b k n}. Then we use the analytic methods to study the mean value properties of the Dirichlet s divisor function dn acting on these two special sets, and obtain two interesting asymptotic formulae for them. That is, we shall prove the following : Theorem. For any real number x, we have the asymptotic formula where dn = n B R p m mx m ζ m+ R p m flog x + +O x +ε m, = p + pm p m m + + p m p + m+ p m p m+ m m+ i= i p m+ i p + m+ p, m This work is supported by N.S.F. of P.R.China

23 4 RESEARCH ON SMARANDACHE PROBLEMS IN NUMBER THEORY II fy is a polynomial of y with degree m = [ k+ ], and ε is any fixed positive number. Theorem. For any real number x, we have the asymptotic formula n C where l = dn = x log x ζl + p l + p p l+ + Ax + +Ox +ε, p [ k ], A is a constant, and ε is any fixed positive number.. Proof of the theorems In this section, we shall complete the proof of the theorems. In fact, let n = p α pα pαs s denotes the factorization of n into prime powers. Then it is clear that b k n = b k p α pα pαs s = b k p α b kp α b kp αs s. That is, b k n is a multiplicative function. So we first study the problem in the case n = p α. If α k, then from the definition of b k n we know that b k n n. Therefore n B. If α k, then b k n = p k α. So combining, and the definition of B and C, we deduce that n B if α [ k+ ], and n C if α [ k ]. Now we prove the theorem and theorem respectively. First let fs = n B dn n s. Then from the definition of b k n and B, the properties of the Dirichlet s divisor function and the Euler product formula [], we have fs = p + dpm p ms + dpm+ p m+s + + m + = p p ms + m + p m+s + = + m + p p ms + m + p ms p s + = ζm+ ms ζ m+ ms p m+ p s i= p s p ms p s + pms p s m + + p s p ms + m+ p s m+ i p ms + m+ p s p mm+ is,

24 On the mean value of the Dirichlet s divisor function in some special sets 5 where ζs is the Riemann zeta-function and m = [ k+ ]. Obviously, we have dn n, n= dn n σ σ, m where σ > m is the real part of s. Therefore by Perron formula [3], with s 0 = 0, b = m, T = x 3 m, we have where n B Rs = p dn = πi + m +it m it ζm+ ms ζ m+ ms Rsxs s ds + Ox m +ε, m+ p ms p s m + + p s p s i= p ms + m+ p s m+ i p mm+ is. To estimate the main term πi m +it m it we move the integral line from s = m ζm+ ms ζ m+ ms Rsxs s ds, ± it to s = m ± it, then the function ζ m+ ms ζ m ms Rsxs s have one m + order pole point at s = m with residue lim s m! = lim s m! m 0 = + lim s m! m m ms m+ ζ m+ Rsx s ms ζ m+ mss m ms m+ ζ m+ Rsx s ms ζ m+ mss + m Rsx ms m+ ζ m+ s ms + lim s m! m m ms m+ ζ m+ ms mx m ζ m+ R m flog x + O x m +ɛ, Rsx s ζ m+ mss ζ m+ mss m + where fy is a polynomial of y with degree k, and ε is any fixed positive number.

25 6 RESEARCH ON SMARANDACHE PROBLEMS IN NUMBER THEORY II So that we can get = m +it πi m it + mx m ζ m+ R m m +it + m +it m it + m +it flog x + Ox m +ɛ. m it m it ζ m+ msx s ζ m+ mss Rsds It is easy to estimate πi and πi πi Therefore, we have where dn = n B R p m = p m +it m +it m it ζm+ msxs ζ m+ mss Rsds ζm+ msxs it ζ m+ mss Rsds m m it m +it ζm+ msxs ζ m+ mss Rsds x m +ε, x m +ε, x m +ε. mx m ζ m+ R p m flog x + +O x +ε m, p m m+ p m p m m + + p m + p + m+ p m i= m+ i p m+ i. This completes the proof of Theorem. For any integer k and any real number s >, let gs = n C dn n s. Then from the Euler product formula [3] we have gs = n C = p dn n s + dp p s + dp p s + + dpl p ms

26 On the mean value of the Dirichlet s divisor function in some special sets 7 = + p p s + 3 p s + + l + p ls = p + p p s + p s + p 3s + + p ls l + p l+s s = ζs p l+s p l + p p l+s s ζ s = l + ps ζl + s p p l+s p s, where l = [ k ]. So by Perron formula [3] and the methods of proving Theorem we can easily get Theorem. References [] F. Smarandache. Only Problems, Not solutions. Chicago: Xiquan Publishing House, 993. [] Apostol T M. Introduction to Analytic Number Theory. New York: Springer- Verlag, 976. [3] Pan Chengdong and Pan Chengbiao. Foundation of Analytic Number Theory. Beijing: Science press, 997. [4] Zhu Weiyi. On the k-power complement and k-power free number sequence. Smarandache Notions Journal, 004, 4: [5] Liu Hongyan and Lou Yuanbing. A Note on the 9-th Smarandache s problem. Smarandache Notions Journal, 004, 4:

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28 A NUMBER THEORETIC FUNCTION AND ITS MEAN VALUE Ren Ganglian Department of Mathematics, Northwest University, Xi an, Shaanxi, P.R.China Abstract Let p and q be two distinct primes, e pqn denote the largest exponent of power pq which divides n. In this paper, we study the properties of thesequence e pqn, and give an interesting asymptotic formula for the mean value e pqn. Keywords: Largest exponent; Asymptotic formula; Mean value.. Introduction Let p and q be two distinct primes, e pq n denote the largest exponent of power pq which divides n. In problem 68 of [], Professor F. Smarandache asked us to study the properties of the sequence e p n. About this problem, some people had studied it, and obtained a series of interesting results See references [4]. In this paper, we use the analytic method to study the properties of the sequence e pq n, and give a sharp asymptotic formula for its mean value e pq n. That is, we shall prove the following: Theorem. Let p and q be two distinct primes, then for any real number x, we have the asymptotic formula e pq n = x pq + O x /+ε, where ε is any fixed positive number.. Proof of the theorem In this section, we shall complete the proof of Theorem. For any complex s, we define the function fs = n= e pq n n s. This work is supported by N.S.F. of P.R.China

29 0 RESEARCH ON SMARANDACHE PROBLEMS IN NUMBER THEORY II Note that the definition of e pq n, and applying the Euler product formula See Theorem.6 of [], we may get fs = = = α=0 α=0 t= n = n,pq= + α=0 t= n = n,pq= α pq αs p ts + α=0 ζs pq s, α p α+t q α n s t= α pq αs α p α q α+t n s + t= n = n,pq= q ts n s n = n,pq= α=0 n = n,pq= n s + α=0 α p α q α n s α pq αs n = n,pq= n s where ζs is the Riemann zeta-function. Obviously, we have e pq n log pq n ln n n= e pq n n σ σ, where σ is the real part of s. Therefore by Parron s formula See reference [3] we can get e pq n n s = b+it ζs + s 0 x s x b 0 πi b it pq s+s 0 s ds + O Bb + σ 0 T { +O x σ 0 Hx min, log x T } { + O x σ 0 HN min, } x x where N is the nearest integer to x, x = x N. Taking s 0 = 0, b = 3, Hx = ln x, Bσ = σ, we have e pq n = πi To estimate the main term πi 3 +it ζs 3 it pq s 3 +it x s s x s pq s ζs 3 it 3 +ε ds + Ox T. s ds,

30 A number theoretic function and its mean value we move the integral line from s = 3 ± it to s = ± it. This time, the function gs = ζs x s pq s s has a simple pole point at s =, and the residue is 3 +it + πi 3 it +it + 3 +it it + +it 3 it it x pq. So we have ζs x s pq s s ds = Taking T = x, and note that +it 3 + it ζs x s πi 3 +it it pq s s ds 3 ζσ + it x 3 pq σ T dσ and πi it ζs +it x 3 +ε T x s pq s = x +ε s ds T we may immediately get the asymptotic formula e pq n = This completes the proof of Theorem. References 0 x pq. ζ + it x pq t dt x +ε, x pq + O x /+ε. [] F. Smarandache. Only Problems, Not solutions. Chicago: Xiquan Publishing House, 993. [] Apostol T M. Introduction to Analytic Number Theory. New York: Springer- Verlag, 976. [3] Pan Chengdong and Pan Chengbiao. Foundation of Analytic Number Theory. Beijing: Science press, 997. [4] Zhang Wenpeng. Reseach on Smarandache problems in number theory. Hexis Publishing House, 004.

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32 ON THE ADDITIVE K-TH POWER COMPLEMENTS Ding Liping Department of Mathematics, Northwest University, Xi an, Shaanxi, P.R.China Abstract Keywords: The main purpose of this paper is using the elementary and analytic methods to study the mean value properties of the additive k-th power complements, and give some interesting asymptotic formulae for it. Additive k-th power complements; Mean value; Asymptotic formula.. Introduction For any positive integer n, the Smarandache k-th power complements b k n is the smallest positive integer such that nb k n is a complete k-th power, see problem 9 of []. Similar to the Smarandache k-th power complements, the additive k-th power complements a k n is defined as follows: a k n is the smallest nonnegative integer such that a k n + n is a perfect k-th power. For example, if k =, we have the additive square complements sequence {a n} n =,, as follows: a = 0, a =, a 3 =, a 4 = 0, a 5 = 4, a 6 = 3, a 7 =, a 8 =, a 9 = 0,. About this problem, many authors have studied it before, and obtained some interesting results. For example, Z.F. Xu [4] studied the mean value properties of the additive k-th power complements, and gave the following: Proposition. For any real number x 3 and fixed positive integer k, we have the asymptotic formula: a k n = k 4k x k + O x k. For any fixed positive integer m, the definition of the arithmetical function δ m n is { max{d N d n, d, m = }, if n 0, δ m n = 0, if n = 0. In this paper, we shall use the elementary and analytic methods to study the mean value properties of the new arithmetical function δ m a k n, and give This work is supported by N.S.F. of P.R.China

33 4 RESEARCH ON SMARANDACHE PROBLEMS IN NUMBER THEORY II an interesting asymptotic formula for it. That is, we shall prove the following conclusion: Theorem. For any real number x 3 and positive integer m, we have the asymptotic formula: δ m a k n = where p m k k x k p m p p + + O x k, if k > 3, denotes the product over all prime divisors p of m, and ɛ is any fixed positive number. Taking k =, 3 in our Theorem, we may immediately deduce the following: Corollary. For any real number x, we have the asymptotic formulae δ m a n = 3 x 3 p p + + O x 5 +ɛ 4 p m and δ m a 3 n = 9 0 x 5 p 3 p + + O x 4 +ɛ 3. p m. Some lemmas To complete the proof of the theorem, we need following Lemmas: Lemma. For any real number x > and positive integer m, we have the asymptotic formula δ m n = x p m p p + + O x 3 +ɛ, where ɛ is any fixed positive number. Proof. Let s = σ+it be a complex number and fs = n= δ mn n. Note that s δ m n n, so it is clear that fs is a Dirichlet series absolutely convergent for Res>, by the Euler product formula [] and the definition of δ m n we get δ m n fs = n n= s = + δ mp p p s + δ mp p s + + δ mp n p ns + = + δ mp p s + δ mp p s + + δ mp n p ns + p m

34 On the additive k-th power complements 5 + δ mp p s + δ mp p s + + δ mp n p ns + p m = + p s + p s pns p m + p p s + p pn + + ps p ns + p m = p m p s p m p s = ζs p s p p s, p m where ζs is the Riemann zeta-function, and p primes. From and Perron s formula [3], we have δ m n = 5 +it p s p πi p s 5 it ζs p m denotes the product over all xs 5 x s ds + O +ɛ T, where ɛ is any fixed positive number. Now we move the integral line in from s = 5 ± it to s = 3 ± it. This time, the function ζs has a simple pole point at s = with residue Hence, we have 5 it πi and = x p m + 3 it p m 5 +it + 5 it p s p p s x p m 3 +it xs s + 5 +it p p it 3 +it ζs p m p s p p s xs s ds p p +. 4 We can easily get the estimate 5 it 3 + +it ζs p s p πi 3 it 5 +it p s xs s ds p m πi 3 it 3 +it ζs p m p s p p s xs 5 +ɛ x T 5 s ds x 3 +ɛ. 6

35 6 RESEARCH ON SMARANDACHE PROBLEMS IN NUMBER THEORY II Taking T = x, combining, 4, 5 and 6 we deduce that δ m n = x p m This completes the proof of Lemma. p p + + Ox 3 +ɛ. 7 Lemma. For any real number x 3 and any nonnegative arithmetical function fn with f0 = 0, we have the asymptotic formula: [ fa k n = x k t= ] n gt fn + O [ n g x k ] fn, k i where [x] denotes the greatest integer not exceeding x and gt = k Proof. See reference [4]. i= t i. 3. Proof of the theorem In this section, we will complete the proof of the theorem. From the definition of δ m a k n, Lemma and Lemma, we have δ m a k n [ ] = x k δ m n + O [ ] δ mn t= n gt n g x k = [ ] x k k t k p p + + Otk 3 + O x k ln x t= p m [ ] if k > 3 x k = k = p m p p + t= t k + Ox k k p k p + x k + Ox k. p m For the cases of k =, 3, we can also prove the results by Lemma. For example, δ m a n

36 On the additive k-th power complements 7 = = [ ] x δ m n + O [ ] δ mn t= n t n g x [ ] x t p p + + Ot 3 +ɛ + O x ln x t= p m = x 3 3 p m p p + + Ox 5 4 +ɛ. This completes the proof of the theorem. References [] F.Smarandache. Only problems, not solutions, Xiquan Publishing House, Chicago, 993. [] Tom M.Apostol. Introduction to Analytic Number Theory. New York: Springer-Verlag, 976. [3] Pan Chengdong and Pan Chengbiao. Foundation of Analytic Number Theory. Beijing: Science Press, 997. [4] Xu Zhefeng. On the additive k-th power complements. Research on Smarandache Problems in Number Theory, Hexis 3-6, 004.

37

38 ON THE SMARANDACHE-RIEMANN ZETA SEQUENCE Li Jie Department of Mathematics, Northwest University, Xi an, Shaanxi, P.R.China Abstract Keywords: The main purpose of this paper is using the elementary method to study the properties of the Smarandache-Riemann zeta sequence, and solve a conjecture posed by Murthy. Smarandache-Riemann zeta sequence; Murthy s conjecture; Elementary method.. Introduction For any positive integer n, we let an, denotes the sum of the base digits of n. That is, if n = a α + a α a s αs with α s > α s > s... > α 0, where a i = 0 or, i =,,..., s, then an, = a i. For any complex number s with Res >, we define Riemann-zeta function as ζs = n s. n= For any positive integer n, let T n be a number such that ζn = πn T n, where π is ratio of the circumference of a circle to its diameter. Then the sequence T = {T n } n= is called the Smarandache-Riemann zeta sequence. In [], Murthy believed that T n is a sequence of integers. Simultaneous, he proposed the following: Conjecture. No two terms of T n are relatively prime. In this paper, we shall prove the following conclusion: Theorem. There exists infinite positive integers n such that T n is not an integer. From this Theorem we know that the Murthy s conjecture is not correct, because there exists infinite positive integers n such that T n is not an integer. i=

39 30 RESEARCH ON SMARANDACHE PROBLEMS IN NUMBER THEORY II. Some simple lemmas Before the proof of the theorem, some simple lemmas will be useful. Lemma. If ord, n! < n, then T n is not an integer, where ord, n! denotes the order of prime in n!. Proof. See reference []. Lemma. For any positive integer n, we have the identity: α n + i= [ ] n i = n an,, where [x] denotes the greatest integer not exceeding x. Proof. From the properties of [x] we know that [ ] n i = = [ a α + a α ] a s αs i s a j αj i, if α k < i α k ; j=k So from this formula we have 0, if i > α s. α n + [ ] n + [ a α + a α ] + + a s αs i= i = i= i s α j s = a j αj k = a j αj j= k= j= s = a j α j s = a j α j a j j= j= = n an, This completes the proof of Lemma. 3. Proof of the theorem In this section, we shall complete the proof of the theorem. From Lemma and the Theorem 3.4 of [3] we know that ord, n! = α n + i= [ ] n i = n an,. Now it is clear that ord, n! < n if and only if n an, < n.

40 On the Smarandache-Riemann zeta sequence 3 That is an, 3. In fact, there exists infinite positive integers n such that an, 3. For example, taking n = α + α + α 3 + α 4 with α 4 > α 3 > α > α 0, then an, 3. For these n, from Lemma we know that T n is not an integer. Since there are infinite positive integers α, α, α 3 and α 4, it means that there exists infinite positive integers n such that an, 3. Therefore, there exists infinite positive integers n such that T n is not an integer. This completes the proof of Theorem. References [] A.Murthy, Some more conjectures on primes and divisors, Smarandache Notions J.00, pp. 3. [] Le Maohua, The Smarandache-Riemann zeta sequence, Smarandache Notions J.4004, pp [3] Tom M. Apostol, Introduction to Analytic Number Theory, New York: Springer-Verlag, 976.

41

42 ON THE PROPERTIES OF THE HEXAGON-NUMBERS Liu Yanni Department of Mathematics, Northwest University, Xi an, Shaanxi, P.R.China Abstract Keywords: In this paper, we shall use the elementary method to study the properties of the hexagon-numbers, and give an interesting identity involving the hexagonnumbers. Hexagon-number; Arithmetical property; Identity.. Introduction For any positive integer m, if n = mm, then we call such an integer n as hexagon-number see reference []. The hexagon-numbers, 6, 5, 8, are closely related to the hexagons. Moreover, the hexagon-numbers are the partial sums of the terms in the arithmetic progression, 5, 9, 3,, 4n +,. For any positive integer n, let m be the largest positive integer that satisfy the inequality mm n < m + m +. Now we define an = mm, and call an as the hexagon-number part of n. In this paper, we shall use the elementary method to study the convergent properties of the Dirichlet s series fs = a n= s, and give an interesting n identity for the case s =. That is, we shall prove the following: For any real number s >, the infinity series fs is conver- Theorem. gent, and f = n= a n = 5 3 π 4 ln.. Proof of Theorem In this section, we shall complete the proof of Theorem. First from the definition of an we know that there exists m + m + mm = 4m +

43 34 RESEARCH ON SMARANDACHE PROBLEMS IN NUMBER THEORY II solutions for the equation an = mm. So we can easy deduce that fs = n= a s n = m= n= an=mm a s n = m= 4m + m s m s. Thus we may conclude that fs is convergent if s >. That is, s >. Now we shall use the elementary method to calculate the exact value of f. Applying the above identity we have f = m= 4m + m m = m m= + m m= 8 mm m= = m m= + m m= 4m m= m= = ζ + ζ 4 ζ 8 mm = 0ζ = 0ζ 4 = 0ζ 4 m= 8 mm m= m m m= m= , 8 mm where ζs is the Riemann zeta-function. From the Taylor s expansion see reference [] for ln + x we know that ln + x = x x + x3 3 x xn n +. Using this identity with x = we may immediately get ln = Combing all the above, and note that ζ = π 6 we may obtain f = 0ζ 4 ln = 5 3 π 4 ln. This completes the proof of Theorem.

44 On the properties of the hexagon-numbers 35 References [] Tom M. Apostol, Introduction to Analytic Number Theory, New York: Springer-Verlag, 976. [] Tom M. Apostol, Mathematical Analysis nd Edition, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 974.

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46 ON THE MEAN VALUE OF THE K-TH POWER PART RESIDUE FUNCTION Ma Jinping Department of Mathematics, Northwest University, Xi an, Shaanxi, P.R.China Abstract Keywords: Similar to the Smarandache k-th power complements, we define the k-th power part residue function f k n is the smallest nonnegative integer such that n f k n is a perfect k-th power. The main purpose of this paper is using the elementary methods to study the mean value properties of f k, and give an n+ interesting asymptotic formula for it. k-th power part residue function; Mean value; Asymptotic formula.. Introduction and results For any positive integer n, the Smarandache k-th power complements b k n is the smallest positive integer such that nb k n is a perfect k-th power see problem 9 of []. Similar to the Smarandache k-th power complements, Xu Zhefeng in [] defined the additive k-th power complements a k n as follows: a k n is the smallest nonnegative integer such that n + a k n is a perfect k-th power. As a generalization of [], we will define the k-th power part residue function f k n as the smallest nonnegative integer such that n f k n is a perfect k-th power. For example, if k =, we have the square part residue sequence {f n} n =,,... as following: 0,,, 0,,, 3, 4, 0,,, 3, 4,5, 6, 0,,,.... Meanwhile, let p be a prime, e p n denotes the largest exponent of power p which divides n. About the relations between e p n and f k n, it seems that none had studied them before, at least we couldn t find any reference about it. In this paper, we use the elementary methods to study the mean value properties of f k n+ and e pf k n, and obtain two sharper asymptotic formulae for them. That is, we will prove the following conclusions: Theorem. For any real number x 3, we have the asymptotic formula f k n + = k k x k ln x + ln x + γ k + x k + Oln x, where γ is the Euler constant.

47 38 RESEARCH ON SMARANDACHE PROBLEMS IN NUMBER THEORY II Theorem. For any real number x 3, we have the asymptotic formula e p f k n = k p x + O p x k.. Some Lemmas To complete the proof of the above theorems, we need following several Lemmas. Lemma. For any real number x, we have e p n = p x + Oln x. Proof. See reference [3]. Lemma. Let hn be a nonnegative arithmetical function with h0 = 0. Then, for any real number x we have the asymptotic formula: where gt = M hf k n = t= n gt k i= hn + O n gm hn, k t i and M = [x k ], [x] denotes the greatest integer not i exceeding x. Proof. For any real number x, let M be a fixed positive integer such that M k x < M + k. Noting that if n pass through the integers in the interval [t k, t + k, then f k n pass through all integers in the interval [0, t + k t k, so we can deduce that hf k n = = = = M t= M t= M t= M t= t k n<t+ k hf k n + n gt n gt n gt hn + M k hn 0 n<x M k hf k n hn + O hn 0 n M+ k M k hn + O hn. n gm

48 On the mean value of the k-th power part residue function 39 This proves the Lemma. Lemma 3. For any real number x >, we have the asymptotic formula n = ln x + γ + O, x where γ is the Euler constant. Proof. See reference [4]. Lemma 4 For any real number x 3, we have the asymptotic formula f k n = k k x k + O x k. Proof. Let hn = n and M = [x k ], then from Lemma and Euler summation formula see reference [4] we obtain M f k n = n + O n t= n gt n gm This proves Lemma 4. = = [x k ] t= k t k + Ox k k k x k + Ox k. 3. Proof of the theorems In this section, we shall complete the proof of Theorems. First we prove Theorem. Let hn = n and M = [x k ], then from Lemma and Lemma 3 we obtain = = M t= M t= f k n + n gt n + + O lnkt k + ln n gm + O n + t + γ + O = k lnm! + ln k + γm + Oln x = k [x k ] ln[x k ] [x k ] +ln k + γ[x k ] + Oln x = k k x k ln x + ln x + γ k + x k + Oln x. + Oln x gt

49 40 RESEARCH ON SMARANDACHE PROBLEMS IN NUMBER THEORY II This completes the proof of Theorem. The proof of Theorem. Note that the definition of e p n, from Lemma and Lemma 4 we have e p f k n = = = = = M e p f k n + e p f k n t k n<t k M k n<x t+ M k t k e p j + O e p f k n M k n<m+ k t= t= j=0 M t + k t k + O ln t + k t k p t= +O e p n 0 n<m+ k M k M + k + Oln km k + O p k p M k + O p M k. Taking M = [x k ], we easily get e p f k n = k p x + O p x k. This completes the proof of Theorem. References gm p [] F.Smarandache. Only Problems, not Solutions, Xiquan Publishing House, Chicago, 993. [] Xu Zhefeng. On the Additive k-power Complements. Research on Smarandache Problems in Number Theory, Hexis 3-6, 004. [3] Lv Chuan. A Number Theoretic Function and its Mean Value. Research on Smarandache Problems in Number Theory, Hexis 33-36, 004. [4] Tom M. Apostol. Introduction to Analytic Number theory. New York: Springer-Verlag, 976. [5] Pan Chengdong and Pan Chengbiao. Foundation of Analytic Number Theory. Beijing: Science Press, 997, P98.

50 ON THE INTEGER PART OF THE M-TH ROOT AND THE K-TH POWER FREE NUMBER Li Zhanhu,. Department of Mathematics, Northwest University, Xi an, Shaanxi, P.R.China. Department of Mathematics, Xianyang Teacher s College, Xianyang, Shaanxi, P.R.China Abstract Keywords: The main purpose of this paper is using the elementary methods to study the mean value properties of a special arithmetical function involving the integer part of the m-th root of an integer and the k-th power free numbers, and give an interesting asymptotic formula for it. Integer part of the m-th root; k-th power free numbers; Asymptotic formula.. Introduction and results Let m and k are two fixed positive integers with k. For any positive integer n, we define arithmetical function b m n as the integer part of the m- th root of n. That is, b m n = [n m ], where [x] denotes the greatest integer x. For example, b =, b =, b 3 =, b 4 =, b 5 =, b 6 =, b 7 =, b 8 =, b 9 = 3,. In reference [], Professor F.Smarandache asked us to study the properties of the sequences {b k n}. About this problem, I do not know whether there exists any progress. But none study the mean value properties of {b k n} over the k-th power free numbers, here we call a positive integer n as k-th power free number, if p k n for any prime p. For convenience, we let A k denotes the set of all k-th power free numbers. In this paper, we shall use the elementary methods to study the mean value properties of b m n over the set A k, and give an interesting formula for it. That is, we shall prove the following result: Theorem. Let m and k are two fixed positive integers with k. Then for any real number x, we have the asymptotic formula b m n = n A k where ζk is the Riemann zeta-function. m ζk m + x m+ m + O x, From this theorem we may immediately deduce the following Corollaries:

51 4 RESEARCH ON SMARANDACHE PROBLEMS IN NUMBER THEORY II Corollary. b n = n A For any real number x, we have the asymptotic formula 4 π x 3 + O x and b 3 n = n A 9 π x O x. Corollary. b 4 n = n A 4 For any real number x, we have the asymptotic formula 64 π 4 x O x and b 5 n = n A 4 75 π 4 x O x.. Some Lemmas Before completing the proof of the theorem, we need the following lemmas. Lemma. Let m and k are two fixed positive integers with k. Then for any real number x, we have the asymptotic formula n m m = m + x m+ µn m + Ox and n k = ζk + O x k+, where µn is the Möbius function, and ζk is the Riemann zeta-function. Proof. Using the Euler s summation formula see Theorem 3. of [] we can easily deduce these conclusions. Lemma. Let an denotes the character function of the k-th power free numbers That is, if n is a k-th power free number, then an = ; if n is not a k-th power free number, then an = 0. Then for any positive integer n, we have the identity an = µd, d k n where µd is the Möbius function. Proof. From the properties of Möbius function we know that for any positive integer n, [ µd =. n] d n Let u k denotes the greatest k-th power divisor of n. That is, n = u k v, where v is a k-th power free number. It is clear that n is a k-th power free number if and only if u =. In this case, µd = µd = µd = µ = = an. d k n d k u k d u If u >, then from the above formula we have µd = [ µd = µd = = 0 = an. u] d k n d k u k d u

52 On the integer part of the m-th root and the k-th power free number 43 This proves Lemma. 3. Proof of the theorem In this section, we shall complete the proof of the theorem. For any real number x and positive integer k, from Lemma and Lemma and the definition of b m n we have b m n n A k = anb m n d k t x = d k n = µd = d k x = = = d k x [ ] µd n m µdd k m µdd k m m m + x m+ m [ ] d k t m t m + O t x/d k d k x t x/d k m m + x m+ x m + O d km+ m d k x m d k x µd d k + O m ζk m + x m+ m + O x. d k x m + O x + O x This completes the proof of Theorem. Now the Corollaries follows from ζ = π /6 and ζ4 = π 4 /90. References d k d k x [] Smarandache F. Only problems, not Solutions. Chicago: Xiquan Publ. House, 993. [] Tom M.Apostol. Introduction to Analytic Number Theory. New York: Springer-Verlag, 976. [3] Zhang Wenpeng. Research on Smarandache problems in number theory. Hexis Publishing House, 004.

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54 A FORMULA FOR SMARANDACHE LCM RATIO SEQUENCE Wang Ting Department of Mathematics, Xi an Science and Arts College, Xi an, Shaanxi, P.R.China Abstract Keywords: In this paper, a reduction formula for Smarandache LCM ratio sequences SLR5 is given. Smarandache LCM ratio sequences; Reduction formula.. Introduction Let x, x,..., x t and [x, x,..., x t ] denote the greatest common divisor and the least common multiple of any postive integers x, x,..., x t respectively. Let r be a positive integer with r >. For any positive integer n, let [n, n +,..., n + r ] T r, n =, [,,...r] then the sequences SLRr = {T r, n} is called Smarandache LCM ratio sequences of degree r. In reference [], Maohua Le studied the properties of SLRr, and gave two reduction formulas for SLR3 and SLR4. In this paper, we study the calculating problem of SLR5, and prove the following conclusion. Theorem. For any postive integer n, we have the calculating formula : T 5, n = Proof of the theorem nn + n + n + 3n + 4, if n 0, 8 mod ; nn + n + n + 3n + 4, if n, 7 mod ; nn + n + n + 3n + 4, if n, 6 mod ; nn + n + n + 3n + 4, if n 3, 5, 9, mod ; nn + n + n + 3n + 4, if n 4 mod ; nn + n + n + 3n + 4, if n 0 mod. To complete the proof of Theorem, we need following several simple Lemmas. Lemma. For any postive integer a and b, we have a, b[a, b] = ab.

55 46 RESEARCH ON SMARANDACHE PROBLEMS IN NUMBER THEORY II Lemma. For any postive integer s with s < t, we have x, x,..., x t = x,..., x s, x s+,..., x t and [x, x,..., x t ] = [[x,..., x s ], [x s+,..., x t ]]. Lemma 3. For any postive integer n, we have T 4, n = { 4 7 nn + n + n + 3, if n, mod 3; nn + n + n + 3, if n 0 mod 3. The proof of Lemma and Lemma can be found in [3], Lemma 3 is proved in reference []. Now we use these Lemmas to complete the proof of our Theorem. In fact, for any positive integer n, from the properties of the least common multiple of any positive integers we know that [n, n +, n +, n + 3, n + 4] = [[n, n +, n +, n + 3], n + 4] = [n, n +, n +, n + 3]n + 4 [n, n +, n +, n + 3], n + 4. Note that [,, 3, 4, 5] = 60, [,, 3, 4] = and 4, if n 0, 4 mod ;, if n, 3, 5, 7, 9 mod ; 6, if n mod ; [n, n +, n +, n + 3], n + 4 =, if n 6, 0 mod ;, if n 8 mod ; 3, if n mod, Combining Lemma 3, we and easily get the theorem. References [] Maohua Le, Two formulas for smarandache LCM ratio sequences, Smarandache Notions Journal, Vol.4, 004, [] F.Smarandache, Only problems, not Solutions, Xiquan Publ. House, Chicago, 993. [3] Tom M.Apostol, Introduction to Analytic Number Theory, Springer- Verlag, New York, Heidelberg Berlin, 976.

56 ON THE M-TH POWER COMPLEMENTS SEQUENCE Yang Hai Research Center for Basic Science, Xi an Jiaotong University, Xi an, Shaanxi, P.R.China Fu Ruiqin School of Science, Xi an Shiyou University, Xi an, Shaanxi, P.R.China Abstract Keywords: The main purpose of this paper is using the elementary and analytic methods to study the asymptotic properties of the m-th power complements sequence, and give several interesting asymptotic formulae for it. M-th power complements; Riemann zeta-function; Asymptotic formula.. Introduction and results For any positive integer n, let b m n denote the m-th power complements sequence. That is, b m n denotes the smallest positive integer such that nb m n is a complete m-th power. In problem 9 of the reference [], Professor F.Smarandache asked us to study the properties of this sequence. About this problem, some authors had studied it before, and obtained some interesting results, see reference [4] and [5]. The main purpose of this paper is using the elementary and analytic methods to study the asymptotic properties of the m- th power complements sequence, and give an interesting asymptotic formula for it. That is, we shall prove the following: Theorem. For any real number x > and any fixed positive integers m and k, we have the asymptotic formula = δ k b m n x p m p m p m + ζm p m p k p k p m+ p + p m + O x 3 +ɛ, This work is supported by N.S.F. of P.R.China 07093

57 48 RESEARCH ON SMARANDACHE PROBLEMS IN NUMBER THEORY II where δ k n defined as following: δ k n = max{d N d n, d, k = }, denotes the product over all prime numbers. p From this theorem, we may immediately get the following: Corollary. Let an be the square complements sequence, then for any real number x > and any fixed positive integer k, we have the asymptotic formula δ k an = 3x p 4 p 3 + p p 3 π p 4 p + p + O x 3 +ɛ. p k Corollary. Let bn be the cubic complements sequence, then for any real number x > and any fixed positive integer k, we have the asymptotic formula δ k bn = x p 6 p 5 + p 3 p 4 ζ3 p 6 p + p 3 +O x 3 +ɛ. p k p k p k. Proof of the theorem In this section, we will complete the proof of Theorem. Let s = σ + it be a complex number and fs = δ k b mn δ k bmn n n= s. Note that, so it is clear that fs is a Dirichlet series absolutely convergent for Res>. By the Euler product formula [] and the multiplicative property of δ k b m n we get fs = = p n= δ k b mn + n s δ k b mp p s + δ k p m p s + + δ k b mp p s + + δ k p ms + δ k p m δ k b mp m p ms + δ k = + p p m+s + + p ms + = p ms + p s p ms p k p k p s ζs p s p m s = ζms p ms + pm s p ms p k

58 On the m-th power complements sequence 49 p s p ms p m s p s p ms, p k where ζs is the Riemann zeta-function and denotes the product over all p prime numbers. From and Perron s formula [3], we have δ k b m n = 5 +it ζs p s p m s πi 5 it ζms p p k ms p s p ms p m s p s p ms xs 3 x s ds + O T p k 3 + pm s +ɛ p ms, 4 where ɛ is any fixed positive number. Now we move the integral line in from s = 5 ± it to s = 3 ± it. This time, the function ζs p s p m s ζms p p k ms + pm s p ms p s p ms p m s p s p ms xs s p k has a simple pole point at s = with residue x p m p m p m + ζm p m p k Hence, we have p k p m+ p + p m. 5 = 5 it πi 3 it + 5 +it 3 + +it + 5 it 5 +it p s p m s 3 it 3 +it + pm s ζs ζms p p k ms p ms 6 p s p ms p m s p s p ms xs s ds p k x p m p m p m + p m+ ζm p m p + p m. 7 p k p k

59 50 RESEARCH ON SMARANDACHE PROBLEMS IN NUMBER THEORY II We can easily get the estimate 5 it fs xs πi 3 it s ds x 5 +ɛ T, 8 and πi πi 3 +it fs xs 5 +it s ds 3 it x 5 +ɛ T, 9 fs xs 3 +it s ds x 3 +ɛ. 0 Taking T = x, combining, 4, 5, 6 and 7 we may deduce that = δ k b m n x p m p m p m + ζm p m p k This completes the proof of Theorem. References p k p m+ p + p m + O x 3 +ɛ. [] Smarandache F. Only problems, not Solutions. Chicago: Xiquan Publ. House, 993. [] Tom M.Apostol. Introduction to Analytic Number Theory. New York: Springer-Verlag, 976. [3] Pan Chengdong and Pan Chengbiao. Foundation of Analytic Number Theory. Beijing: Science Press, 997. [4] Zhu Weiyi. On the k-power complements and k-power free number sequence. Smarandache Notions Journal, 004, 4: [5] Liu Hongyan and Lou Yuanbing. A note on the 9-th Smarandache s problem. Smarandache Notions Journal, 004, 4:

60 ON THE SMARANDACHE CEIL FUNCTION AND THE DIRICHLET DIVISOR FUNCTION Ren Dongmei Research Center for Basic Science, Xi an Jiaotong University, Xi an, Shaanxi, P.R.China Abstract Keywords: The main purpose of this paper is using the elementary methods to study the mean value properties of the composite function involving Dirichlet divisor function and Smarandache ceil function, and give an interesting asymptotic formula for it. Smarandache Ceil function; Dirichlet divisor function; Mean value; Asymptotic formula.. Introduction For a fixed positive integer k and any positive integer n, the Smarandache ceil function S k n is defined as following: S k n = min{m N : n m k }. This function was introduced by Professor Smarandeche see reference []. About this function, many scholars studied its properties see reference [] and [3]. In reference [], Ibstedt presented the following property: a, b Na, b = = S k ab = S k as k b. That is, S k n is a multiplicative function. In this paper, we use elementary methods to study the mean value properties of the composite function involving dn and S k n, and give an interesting asymptotic formula for it. That is, we shall prove the following: Theorem. Let k be a given positive integer with k. Then for any real number x, we have the asymptotic formula: ds k n = 6ζkx ln x π p p k + p k + Cx + Ox +ε. where C is a computable constant, and ε is any fixed positive number. Taking k = and k = 4 in Theorem, noting that: ζ = π π4, ζ4 = 6 90,

61 5 RESEARCH ON SMARANDACHE PROBLEMS IN NUMBER THEORY II we may immediately deduce the following: Corollary. For any real number x, we have the asymptotic formula: ds n = x ln x p p + C x + Ox +ε, + p ds 4 n = π x ln x 5 p p 4 + p 3 + C x + Ox +ε, where C, C are computable constants. If k =, then function S k n turns into the identical transformation and the composite function ds k n turns into dn. We have the following asymptotic formula: ds n = dn = x ln x + γ x + Ox, where γ is the Euler s Constant.. Proof of the theorem In this section, we shall complete the proof of the theorem. Let fs = n= ds k n n s. From the Euler product formula see reference [5] and the multiplicative property of S k n, we have fs = p + ds kp p s + ds kp p s + + dp p s = p = + p p s + + p ks + 3 p k+s + = + p ks p p p s + 3 p k+s + s = ζs + p p s + p ks + + pks = ζsζks p p ks + p s, = ζsζks ζs ζs + + dp p ks + dp p k+s + + dp p ks + p p ks + p k s

62 On the Smarandache ceil function and the Dirichlet divisor function 53 where ζs is the Riemann zeta-function. It is obvious that ds k n ds k n n, n σ σ, n= where σ > is the real part of s. By Perron formula see reference [4], let s 0 = 0, b = 3, T = x, then we have ds k n = 3 +it ζ sζks πi 3 it ζs where Rs = p p ks + p k s Now we calculate the term πi 3 +it ζ sζks 3 it ζs Rs xs s ds + Ox +ε,, and ε is any fixed positive number. Rs xs s ds. We move the integral line from 3 ± it to ± it. Then the function ζ sζks Rs xs ζs s has a second order pole at s = with residue lim s ζ sζks Rs xs s ζs s = lim s ζ sζks x s Rs s ζs s +s ζ sζks Rs sxs ln x x s ζs s 6ζkx lnx = π p k + p k + Cx, p where C is a computable constant. So we can obtain 3 +it + +it + it 3 + it ζ sζks Rs xs πi 3 it 3 +it +it it ζs s ds 6ζkx lnx = π p k + p k + Cx. Taking T = x, we have the estimate +it πi << x 3 +ε T p + 3 +it = x +ε 3 it it ζ sζks Rs xs ζs s ds

63 54 RESEARCH ON SMARANDACHE PROBLEMS IN NUMBER THEORY II and it ζ sζks Rs xs πi +it ζs s ds << x +ε. So we may immediately get the asymptotic formula ds k n = 6ζkx lnx π This completes the proof of Theorem. References p p k + p k + Cx + Ox +ε. [] F. Smarandache. Only Problems, Not solutions. Chicago: Xiquan Publishing House, 993. [] Ibstedt. Surfining On the Ocean of Number-A New Smarandache Notions and Similar Topics. New Mexico: Erhus University Press. [3] Sabin Tabirca and Tatiana Tairca. Some new results concerning the Smarandache ceil function, Smarandache Notions, book series Vol. 3, 00, [4] Pan Chengdong and Pan Chengbiao. Foundation of Analytic Number Theory. Beijing: Science Press, 999. [5] Tom M, Apostol. Introduction to Analytic Number Theory. New York: Springer-Verlag, 976.

64 ON A DUAL FUNCTION OF THE SMARANDACHE CEIL FUNCTION Lu Yaming Research Center for Basic Science, Xi an Jiaotong University, Xi an, Shaanxi, P.R.China Abstract Keywords: Let n and k are any two positive integers, S k n denotes the largest positive integer x satisfying x k n. The main purpose of this paper is using the elementary methods to study the mean value properties of the arithmetical function ds k n, where dn is the Dirichlet divisor function, and give an interesting asymptotic formula for it. Arithmetical function; Mean value; Asymptotic formula.. Introduction and results Let k be a fixed positive integer. For any positive integer n, the Smarandache ceil function of order k is denoted by S k n and has the following definition: S k n = min{x N : n x k }. This arithmetical function is a multiplicative function, and has many interesting properties, so it had be studied by many people, see reference []. Now we introduce a dual function of S k n as follows: S k n = max{x N : x k n}. It is clear that S k n is also a multiplicative function, but about its arithmetical properties, we know very little at present. In this paper, we use the elementary methods to study the mean value properties of ds k n, where dn is the Dirichlet divisor function, and obtain a sharper asymptotic formula for it. That is, we will prove the following: Theorem. Let k be a fixed integer. Then for any real number x >, we have the asymptotic formula ds n = x ln x + γ x + O x 3 and ds k n = ζkx + ζ k x k + O x k+,

65 56 RESEARCH ON SMARANDACHE PROBLEMS IN NUMBER THEORY II where γ is the Euler constant, and ζn is the Riemann zeta-function. From this Theorem we may immediately deduce the following Corollary. For any real number x >, we have the asymptotic formula ds n = π 6 x + ζ x + O x 3. Corollary. For any real number x >, we have the asymptotic formula ds 4 n = π4 90 x + ζ x 4 + O x Proof of the theorem In this section, we will give the proof of the theorem. The following Lemma is necessary. Lemma. If x and s 0, s, we have where n s = x s s + ζs + Ox s, ζs = + s s t [t] dt. ts+ Proof. This Lemma can be easily proved by using the Euler summation formula, see Theorem 3.b of [3]. Now we come to the proof of the theorem. It is obvious that S n = n, and this deduces the first part of the theorem immediately by the classical result on Dirichlet divisor problem, see reference [4]. Now assume k, we have ds k n =, d S k n from the definition of S k n we know that d S k n d k n, hence ds k n = =. d k n d k l x Let δ = x k+, applying the above formula and Theorem 3.7 of [3] we have ds k n = d k l x

66 On a dual function of the Smarandache ceil function 57 = + d k δ k l x/d k l δ d k x/l d k δ k l δ = + [δ] d δ l x/d k l δ d k x/l = [ ] x d k + [ x ] /k δ δ{δ} + {δ}. l d δ l δ Using the above Lemma we get ds k n = x d k + x/k l d δ l δ /k + Oδ δ + Oδ δ k δ = x k + ζk + Oδ k + x /k k + ζ + Oδ k k k δ + Oδ. Notice that x = δ k+, we have ds k n δ δ = + ζkx + k + ζ k = ζkx + ζ x k + Oδ k = ζkx + ζ This completes the proof of Theorem. References k x k + Ox k+. k x k δ + Oδ [] F. Smarandache. Only Problems, Not solutions. Chicago: Xiquan Publishing House, 993. [] Sabin Tabirca and Tatiana Tairca. Some new results concerning the Smarandache ceil function, Smarandache Notions, book series Vol. 3, 00, [3] Tom M, Apostol. Introduction to Analytic Number Theory. New York: Springer-Verlag. 976 [4] Pan Chengdong and Pan Chengbiao. Foundation of Analytic Number Theory. Beijing: Science Press, 999.

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68 ON THE LARGEST M-TH POWER NOT EXCEEDING N Liu Duansen and Li Junzhuang Institute of Mathematics, Shangluo Teacher s College, Shangluo, Shaanxi, P.R.China Abstract Keywords: In this paper, some elementary methods are used to study the properties of the largest m-th power not exceeding n, and give an identity about it. The largest m-th power not exceding n ; Dirichlet series; Identity.. Introduction and results Let n is a positive integer. It is clear that there exists an integer k such that k m n < k + m. Now we define b m n = k m. That is, b m n is the largest m-th power not exceeding n. In problems 40 and 4 of reference [], Professor F.Smarandache asked us to study the properties of the sequences {b n} and {b 3 n}. About these problems, some people had studied them, and obtained some interesting results, see references [] and [3]. In this paper, we using the elementary methods to study the convergent properties of the Dirichlet series fs = b s, and give an interesting identity. That is, we shall prove the following: n= m n Theorem. Let m be a fixed positive integer. Then for any real number s >, the Dirichlet series fs is convergent and fs = Cmζms m++c mζms m++ +Cm m ζms +ζms, where Cm n m! = n!m n!, and ζs is the Riemann zeta-function. From this theorem we may immediately deduce the following: This work is supported by N.S.F. of P.R.China07093 and the Education Department Foundation of Shannxi Province04JK3

69 60 RESEARCH ON SMARANDACHE PROBLEMS IN NUMBER THEORY II Corollary. Taking m = and s = 3/ or m = s = in the above Theorem, then we have the identities 3 b n = π 3 + ζ3 and π4 b = ζ3 + n 90. n= Corollary. Taking m = 3 and s = or m = and s = 3 in the above theorem, we have the identities n= n= b 3 n = π4 π6 + 3ζ and n= π6 b 3 = ζ5 + n Proof of the theorem In this section, we shall complete the proof of the theorem. For any positive integer n, let b m n = k m. It is clear that there are exactly k + m k m integer n such that b m n = k m. So we may get fs = = n= k= b s mn = k= k + m k m k ms C mk m + C mk m + + C m m k + k ms. From the integral criterion, we know that fs is convergent if ms m >. That is, s >. If s >, note that ζs =, we have ns fs = Cmζms m + + Cmζms m Cm m ζms + ζms. This completes the proof of Theorem. It s easy to compute that f = Cmζm + + Cmζm Cm m ζm + ζm, f3 = Cmζm + + Cmζm Cm m ζ3m + ζ3m. Now the corollaries follows from ζ = π /6, ζ4 = π 4 /90, ζ6 = π 6 /945 see reference [4] and the above theorem. References [] F. Smarandache. Only Problems, Not solutions. Chicago: Xiquan Publishing House, 993. n=

70 On the largest m-th power not exceeding n 6 [] Zhang Wenpeng. On the cube part sequence of a positive integer. Journal of Xian Yang Teacher s College, 003, 8: 5-7. [3] Zheng Jianfeng. On the inferior and superior k-th power part of a positive integer and divisor function. Smarandache Notions Journal, 004, 4: [4] Tom M. Apostol. Introduction to Analytic Number Theory. New York: Springer-Verlag, 976.

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72 ON THE SMARANDACHE BACK CONCATENATED ODD SEQUENCES Li Junzhuang and Wang Nianliang Institute of Mathematics, Shangluo Teacher s College Shangluo, Shaanxi, P.R.China Abstract The main purpose of this paper is to study the arithmetical properties of the Smarandache back concatenated odd sequences, and give several simple properties involving the recursion formula and exact expressions for the general term. Keywords: The Smarandache back concatenated odd sequence; Arithmetical properties; Recursion formula.. Introduction and main results The famous Smarandache back concatenated odd sequence {b n } is defined as following, 3, 53, 753, 9753, 9753, 39753, , ,. In problem 3 of reference [], Professor Mihaly Bencze and Lucian Tutescu asked us to study the arithmetical properties about this sequence. It is interesting for us to study this problem. But it s a pity that none had studied it before. At least we haven t seen such a paper yet. In this paper, we shall use the elementary methods to study the arithmetical properties of the Smarandache back concatenated odd sequences, and give several simple properties involving the recursion formula, exact expressions for the general term, and so on. That is, we shall prove the following: Theorem. Let n be any positive integer with n 3 has k digits. Then for the Smarandache back concatenated odd sequences {b n }, we have the following recursion formula where b =. b n = b n + n 0 0 0k 8 +k n, This work is supported by N.S.F. of P.R.China 07093and the Education Department Foundation of Shannxi Province04JK3

73 64 RESEARCH ON SMARANDACHE PROBLEMS IN NUMBER THEORY II Theorem. Let n 3 has k digits. Then the b n -th term in the Smarandache back concatenated odd sequences {b n } has 0 0k 8 +k n digits. n Theorem 3. Let S km,n = i 0 ki, then we have the i=m following exact expression for the general term. That is, k {}}{ b n = +S, S 7, S 35, Skm,n. Theorem 4. Let S 50 denotes the summation for the first 50 terms in the Smarandache back concatenated odd sequences {b n }, then we have S 50 = Some lemmas Lemma. Let k, m, n are positive integers with m n, then we have S km,n = m 0km 0 k + 0km [ 0 kn m ] n 0kn 0 k + 0 k. Proof. From the definition of S km,n, we have S km,n = n i 0 ki i=m = m 0 km + m + 0 km + + n 0 kn and 0 k S km,n = m 0 km +m+ 0 km+ + +n 0 kn. Thus we can get 0 k S km,n = m 0 km + [0 km + 0 km+

74 On the Smarandache back concatenated odd sequences 65 So we have kn ] n 0 kn = m 0 km + 0km [ 0 kn m ] 0 k n 0 kn. S km,n = m 0km 0 k + 0km [ 0 kn m ] n 0kn 0 k + 0 k. This proves Lemma. n Lemma. Let S kn,m = i 0 ki. Then for any positive integers k, m, n, we have i=m S km,n = m 0 km m + 0 k m+ 0 k + 0 k + 0 km+ [ 0 kn m ] 0 k 3 n 0 k n+ 0 k n 0 kn+ 0 k. Proof. From the definition of S km,n, we have S km,n = m 0 k m + m + 0 k m+ + m + 0 k m+ + + [m + n m] 0 k[m+n m] and 0 k S km,n = m 0 k m+ + m + 0 k m+ + + [m + n m] 0 k[m+n m+]. Thus we can get 0 k S km,n = m 0 k m + m + 0 k m+ + m k m+ + + {[m + n m] } 0 k[m+n m] [m + n m] 0 k[m+n m]+.

75 66 RESEARCH ON SMARANDACHE PROBLEMS IN NUMBER THEORY II So we have S km,n = m 0 km 0 k + = m 0 km 0 k + This proves Lemma. Lemma 3. Let S kn,m = k, m, n, we have n i=m+ i 0 k i 0 k + n 0 kn+ 0 k m + 0 k m+ 0 k + 0 km+ [ 0 kn m ] 0 k 3 n 0 k n+ 0 k n 0 kn+ 0 k. n i=m i 0 ki. Then for any positive integers S km,n = m 0 km 0 k + 0 km+ [ 0 kn m ] 0 k n 0 kn+ 0 k. and Proof. From the definition of S km,n, we have S km,n = m 0 km + m + 0 km+ + + [m + n m] 0 k[m+n m] 0 k S km,n = m 0 km+ + m + 0 km+ Thus we can get + + [m + n m] 0 k[m+n m+]. 0 k S km,n = m 0 km + 0 km k[m+n m] So we have [m + n m] 0 k[m+n m+]. S km,n = m 0 km 0 k + 0 km+ [ 0 kn m ] 0 k n 0 kn+ 0 k. This proves Lemma 3.

76 On the Smarandache back concatenated odd sequences Proof of the theorems In this section, we shall use the above lemmas to complete the proof of the theorems. First we use mathematical induction to prove Theorem. i If n =, 3, it is obvious that Theorem is true. ii Assuming that Theorem is true for n = m. That is, the following recursion formula b m = b m + m 0 0 0k 8 +k m is true for the Smarandache back concatenated odd sequence {b n }. Comparing the difference between b m and b m, we know that b m has 0 0 k 8 + k m digits. If n = m +, we can divide it into two cases: i If m still has k digits, then we know that b m has [ 0 0k 8 + k m + k] digits. Comparing the difference between b m+ and b m, we may immediately deduce that That is, b m+ b m m + = 0 0 0k 8 +k m +k. b m+ = b m + m k 8 +k m. ii If m has k + digits, then we know that b m has [ 0 0k 8 + k m + k + ] digits. Recall that m 3 has k digits and m has k + digits, which exist only in the case that m 3 = 0 k, m = 0 k +. That is, m = 0k +. Therefore, we have 0 0 k + k m + k + = 8 = 0 0k + k 0k + k k+ + k + m 8 Comparing the difference between b m+ and b m, we can deduce that That is, b m+ b m m + = 0 0 0k 8 +k m +k+ = 0 0 0k+ 8 +k+ m. b m+ = b m + m k+ 8 +k+ m. Combining i and ii, Theorem is true for any positive integer n. This completes the proof of Theorem. By using the result of Theorem, and note that the difference between b n and b n, we can immediately get the result of Theorem.

77 68 RESEARCH ON SMARANDACHE PROBLEMS IN NUMBER THEORY II By Theorem, when n 3 has k digits, let m 3 be the least positive integer which has k odd digits here m = when k =, and m = 0k +. Then we have b n = m 0 0 0k 8 +k m + + n 0 0 0k 8 +k n S, S 7, S 35, k 8 S km,n S, S 7, S 35, k 8 S km,n = + S, S 7, S 35,50 k {}}{ Skm,n. Applying the result of Lemma, we can deduce Theorem 3. Now we come to prove Theorem 4. By using the result of Theorem, we have S 50 = b + b + b b 49 + b 50 =

78 On the Smarandache back concatenated odd sequences = 50 [ 5 50 ] [ 5 49 ] [ 5 48 ] [ ] [ 5 45 ] [ ] [ 5 ] [ 5 ] = 0 [ ] [ = i=45 44 i= ] i i i i

79 70 RESEARCH ON SMARANDACHE PROBLEMS IN NUMBER THEORY II 50 i=45 44 i= = i i i i 50 i=45 50 i=45 i 0 i i 0 i i= 44 i= Applying the results of Lemma and Lemma 3, we can get S 50 = i 0 i i 0 i This completes the proof of Theorem 4. References [] Mihaly Bencze and Lucian Tutescu Eds.. Some notions and questions in number theory. Vol. smarandache/snaqint.txt. [] F.Smarandache, Only problems, not Solutions, Xiquan Publ. House, Chicago, 993. [3] Tom M.Apostol, Introduction to Analytic Number Theory, Springer- Verlag, New York, Heidelberg Berlin, 976.

80 ON THE ADDITIVE HEXAGON NUMBERS COMPLEMENTS Li chao and Yang Cundian Institute of Mathematics, Shangluo Teacher s College, Shangluo, Shaanxi, P.R.China Abstract Keywords: In this paper, similar to the Smarandache k-th power complements, we defined the hexagon numbers complements. Using the elementary method, we studied the mean value properties of the additive hexagon numbers complements, and obtained some interesting asymptotic formulae for it. Additive hexagon numbers complements; Mean value; Asymptotic formula.. Introduction and results Let n be a positive integer. If there exists a positive integer m such that n = mm, then we call n as a hexagon number. For any positive integer n, the Smarandache k-th power complements b k n is the smallest positive integer such that nb k n is complete k-th power, see problem 9 of []. Similar to the Smarandache k-th power complements, we define the additive hexagon numbers complements an as follows: an is the smallest nonnegative integer such that an + n is a hexagon number. For example, if n =,, 5, we have the additive hexagon number sequences {an} n =,, 5 as follows: 0, 4, 3,,, 0, 8, 7, 6, 5, 4, 3,,, 0. In this paper, we study the mean value properties of the composite arithmetic function dan where dn is the Dirichlet divisor function, and give some interesting asymptotic formulae for it. That is, we shall prove the following conclusions: Theorem. For any real number x 3, we have the asymptotic formula: an = 3 x 3 + Ox. Theorem. For any real number x 3, we have the asymptotic formula: dan = 3 x log x + log + γ x + Ox 3, This work is supported by N.S.F. of P.R.China07093 and the Education Department Foundation of Shannxi Province04JK3

81 7 RESEARCH ON SMARANDACHE PROBLEMS IN NUMBER THEORY II where γ is the Euler constant.. Some Lemmas Before the proof of the theorems, some Lemmas will be useful. Lemma. For any real number x 3, we have the asymptotic formula: dx = x ln x + γ x + O x 3, where γ is the Euler constant. Proof. See reference []. Lemma. For any real number x 3 and any nonnegative arithmetical function fn with f0 = 0, we have the asymptotic formula: [ ] x fan = m= i 4m fi + O i 4 [ x ] fi, where [x] denotes the greatest integer less than or equal to x. Proof. For any real number x, let M be a fixed positive integer such that MM x < M + M +. Noting that if n pass through the integers in the interval [mm, m + m +, then an pass through the integers in the interval [0, 4m] and f0 = 0, we can write fan = fan + fan M = [ x = n MM M m= i 4m fi + O MM < i x MM fi, Since x ] MM < M + M + MM = 4M + and, we have [ ] x fan = m= i 4m fi + O i 4 [ x ] fi.

82 On the additive hexagon numbers complements 73 This proves Lemma. Note: This Lemma is very useful. Because if we have the mean value formula of fn, then from this Lemma, we can easily get the mean value formula of fan. 3. Proof of the theorems In this section, we will complete the proof of the theorems. First, we prove Theorem. From the definition of an, and the Euler summation formula see [3], we have an = an + an = = n MM MM < M i + i m= i 4m i x MM M 4M m4m + + O m= = 4 3 MM + M + + OM = 3 x 3 + Ox. This proves Theorem. Now we prove Theorem. From Lemma, Lemma and the Abel s identity see [3], we have dan = = M m= i 4m M m= di + O di i 4M 4m log 4m + γ 4m + O 4m 3 +O 4M log 4M + γ 4M + O 4M 3 = 8 log + 4γ m + 4 m log m +O m M m 3 m M m M + O 4M log 4M = 8 log + 4γ M + OM

83 74 RESEARCH ON SMARANDACHE PROBLEMS IN NUMBER THEORY II +4 M log M 4 M + OM log M + OM 4 3 = M log M + 4 log + γ M + OM 4 3 = 3 x log x + log + γ x + Ox 3. This completes the proof of the theorems. References [] F.Smaradache. Only problems, not solutions, Xiquan Publishing House, Chicago, 993. [] H.Iwaniec and C.J.Mozzochi. On the divisor and circle problems. Journal of Number theory, Vol. 8988, [3] Tom M. Apostol. Introduction to Analytic Number Theory, Springer- Verlag, New York, 976, pp54 and pp77.

84 ON THE MEAN VALUE OF A NEW ARITHMETICAL FUNCTION Yang Cundian and Liu Duansen Institute of Mathematics, Shangluo Teacher s College, Shangluo, Shaanxi, P.R.China zzz@zzz.com Abstract Keywords: The main purpose of this paper is using the elementary methods to study the mean value properties of a new arithmetical function σsn, and give an interesting mean value formula for it. Smarandache function; Mean value; Asymptotic formula.. Introduction and results For any positive integer n, the Smarandache function Sn is defined as follows: Sn = min{m N : n m!}. This function was introduced by American-Romanian number theorist Professor F.Smarandache, see reference []. About its arithmetical properties, many scholars had studied it, and obtained some interesting conclusions, see reference [] and [3]. In this paper, we shall use the elementary methods to study the mean value properties of a new arithmetical function σsn, where σn = d, and d n give an interesting mean value formula for it. That is, we shall prove the following: Theorem. For any real number x 3, we have the asymptotic formula σsn = π x x ln x + O ln. x This work is supported by N.S.F. of P.R.China07093 and the Education Department Foundation of Shannxi Province04JK3

85 76 RESEARCH ON SMARANDACHE PROBLEMS IN NUMBER THEORY II. Some simple lemmas To complete the proof of Theorem, we need some simple Lemmas. For convenience, we denotes the greatest prime divisor of n by pn. Then we have Lemma. If n is a square free number or pn > n, then Sn = pn. Proof. If n be a square-free number, let n = p p p r pn, then p i < pn for i =,,, r. Thus p i pn!, i =,,, r. So n pn!, but pn pn! and n pn!. In this case, we have Sn = pn; If pn > n, then p n n. Let n = p α pα pαr r pn, so we have It is clear that p α pα pαr r < n < pn. p α i i pn!, i =,,, r. So n pn!, but pn pn!, in this case, we can also deduce that Sn = pn. This proves Lemma. Lemma. Let p be a prime, then we have the asymptotic formula p = x x ln x + O x p x ln. x Proof. Let πx denotes the number of the primes up to x. Noting that see [4] πx = x x ln x + O ln, x from the Abel s summation formula [5], we have This proves Lemma. x p x p = πxx π x x = x ln x = x ln x + O x x x ln x + O ln x x ln. x x πtdt

86 On the mean value of a new arithmetical function Proof of the theorem In this section, we shall complete the proof of Theorem. First we define the sets A and B as following: A = {n n x, pn n} and B = {n n x, pn > n}. Note that Sn pn ln n and σn n ln ln n, we have the estimate Sn ln ln Sn σsn n A n A 3 n ln n ln ln n x ln x ln ln x. Then using the above Lemmas we may immediately get σsn = σpn = σp n B pn> n = = n x = π n ln x x p x n x n ln x + n x x ln x + O Combining and we obtain n x n p x n p + + O ln x ln x n x p n x n p x x n ln x n σsn = σsn + n A n B = π x x ln x + O ln. x This completes the proof of Theorem. References + Ox 3 ln x. σsn [] F. Smarandache. Only problems, Not solutions. Chicago: Xiquan Publ. House, 993. [] Li Hailong and Zhao Xiaopeng. Research on Smarandache Problems In Number Theory. Hexis, 004, 9-. [3] Mark Farris and Patrick Mitchell. Bounding the Smarandache function. Smarandache Notions Journal, Vol 300, [4] Tom M.Apostol. Introduction to Analytic Number Theory. Springer- Verlag: New York, 976.

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88 ON THE MEAN VALUE OF A NEW ARITHMETICAL FUNCTION Zhao Jian and Li Chao Institute of Mathematics, Shangluo Teacher s College, Shangluo, Shaanxi, P.R.China zzz@zzz.com Abstract Keywords: The main purpose of this paper is using the elementary methods to study the convergent properties of a new Dirichlet s series involving the triangular numbers, and give an interesting identity for it. Dirichlet s series; Convergence; Identity.. Introduction and results For any positive integer m, it is clear that mm + / is a positive integer, and we call it as the triangular number, because there are close relations between these numbers and the geometry. Now for any positive integer n, we define arithmetical function cn as follows: cn = max {mm + / : mm + / n, m N}. That is, cn is the greatest triangular number n. From the definition of cn we can easily deduce that c =, c =, c3 = 3, c4 = 3, c5 = 3, c6 = 6, c7 = 6, c8 = 6, c9 = 6, c0 = 0,. About this function, it seems that none had studied it before, even we do not know its arithmetical properties. In this paper, we introduce a new Dirichlet s series fs involving the sequences {cn}, i.e., fs = n= c s n. Then we using the elementary methods to study the convergent properties of fs, and obtain an interesting identity. That is, we shall prove the following result: Theorem Let s be any positive real number. Then the Dirichlet s series fs is convergent if and only if s >. Especially for s =, 3 and 4, we have This work is supported by N.S.F. of P.R.China07093 and the Education Department Foundation of Shannxi Province04JK3

89 80 RESEARCH ON SMARANDACHE PROBLEMS IN NUMBER THEORY II the identities: n= n= n= c n = π 3 4; c 3 n = 8ζ3 4π + 3; c 4 n = π ζ π 384; 6f3 + f4 = π π 9. where ζk is the Riemann zeta-function.. Proof of the theorem In this section, we will complete the proof of the theorem. It is easy to see that if mm+ n < m+m+, then cn = mm+. So the same numbers mm+ repeated m+m+ mm+ {cn}. Hence, we can write fs = = = n= m= m= = m + times in the sequences c s n m + mm+ s s m s m + s. It is clear that fs is convergent if s >, divergent if s. Specially if s =, we can write f = 4 m m= m + = 4 m m + m + m= = 4ζ 4. Using the same method, we can also obtain f3 = 8ζ3 4ζ + 3; f4 = 6ζ4 48ζ3 + 30ζ 384. Now the theorem follows from the identities see reference [] ζ = π /6 and ζ4 = π 4 /90. This complete the proof of the theorem.

90 On the mean value of a new arithmetical function 8 Note: In fact, for any positive integer s, using our methods we can express fs as the Riemann zeta-function. References [] Smarandache F. Only problems, not Solutions. Chicago: Xiquan Publ. House, 993. [] Tom M.Apostol. Introduction to Analytic Number Theory. New York: Springer-Verlag, 976.

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92 ON THE SECOND CLASS PSEUDO-MULTIPLES OF 5 SEQUENCES Gao Nan School of Science, Xi an Shiyou University, Xi an, Shaanxi, P.R.China Abstract Keywords: The main purpose of this paper is to study the mean value properties of the second class pseudo-multiples of 5 sequences, and give an interesting asymptotic formula for it. Second class pseudo-multiples of 5 sequences; Mean value; Asymptotic formula.. Introduction and results A positive integer is called pseudo-multiple of 5 if some permutation of its digits is a multiple of 5, including the identity permutation. In reference [], Professor F.Smarandache asked us to study the properties of the pseudomultiple of 5 sequences. About this problem, Wang Xiaoying [] had studied it, and obtained some interesting results. Now, we define the second class pseudo-multiples of 5 numbers as following: A positive integer is called the second class pseudo-multiple of 5 if it is not a multiple of 5, but its some permutation of its digits is a multiple of 5. For convenience, let A denotes the set of all pseudo-multiple of 5 sequences, and B denotes the set of all second class pseudo-multiple of 5 sequences. In this paper, we shall use the elementary methods to study the mean value properties of the second class pseudo-multiple of 5 sequences, and obtain an interesting asymptotic formula for it. That is, we shall prove the following: Theorem. For any real number x, we have the asymptotic formula n B dn = 6 5 x ln x + γ + ln 5 + O x ln 8 +ε ln 0, where dn is the Dirichlet s divisor function, γ is the Euler constant, and ε denotes any fixed positive number.

93 84 RESEARCH ON SMARANDACHE PROBLEMS IN NUMBER THEORY II. Some Lemmas To complete the proof of the above Theorem, we need the following two Lemmas: Lemma. Let q be a fixed prime or q =. Then for any real number x, we have the asymptotic formula dqn = x ln x + γ + ln q + O x +ɛ, q q where γ is the Euler constant, and ɛ denotes any fixed positive number. Proof. If q =, then from Theorem 3.3 of [3] we know that Lemma is dqn correct. Now for any prime q and real number s >, let fs = n n= s. Note that dn is a multiplicative function, so by the Euler product formula [3] we have fs = = = = n= α=0 dqn n s = + α q sα + q s p p,q= = ζ s α=0 n = n,q= n = n,q= dn n s q s dq α+ n q sα n s + dp p s + dp p s + + dpk p ks + + q s q s. q s ζ s q s Then from this identity and the Perron s formula [4] we may immediately deduce the asymptotic formula dqn = x ln x + γ + ln q + O x +ɛ. q p This completes the proof of Lemma.

94 On the second class pseudo-multiples of 5 sequences 85 Lemma. For any real number x, we have the asymptotic formula dn = x ln x + γ x + O x ln 8 +ε ln 0. n A Proof. For any real number x, it is clear that there exists a nonnegative integer k such that 0 k x < 0 k+. That is, k log x < k +. According to the definition of A we know that the largest number of digits x not in A is 8 k+. In fact, there are 8 one digit, they are,, 3, 4, 6, 7, 8, 9; There are 8 two digits ; ; The number of k digits are 8 k. Since 8 k 8 log x = x ln 8 ln 0, we have n A k 8k k 64 7 x ln 8 ln 0. Note that dn n ε and ln 8 n A This proves Lemma. ln 0 > 3. Proof of the theorem dn = dn n A, applying Lemma with q = we have dn = dn + O = dn + O x ε n A x ln 8 +ε ln 0 = x ln x + γ x + O x ln 8 +ε ln 0. In this section, we complete the proof of Theorem. From the definition of set A and set B, we know the relationship between them: A B={multiples of 5}. Combining the above Lemmas we may immediately get dn d5n n B dn = n A 5 = x ln x + γ x + O 9 5 x = 6 5 x x ln 8 ln 0 +ε ln x ln 5 + γ + ln 5 9 ln x + γ + ln 5 + O x ln 8 ln 0 +ε.

95 86 RESEARCH ON SMARANDACHE PROBLEMS IN NUMBER THEORY II This completes the proof of Theorem. Notes: In fact, we may use the similar method to study other arithmetical functions on the pseudo-multiples of 5 sequences and the second class pseudomultiples of 5 sequences. References [] F.Smarandache. Only problems, not solutions, Xiquan Publishing House, Chicago, 993. [] Wang Xiaoying. On the Smarandache Pseudo-multiples of 5 Sequence, Hexis, 004, 7-9. [3] Tom M. Apostol. Introduction to Analytic Number Theory, New York, 976. [4] Pan Chenddong and Pan Chengbiao. Elements of the Analytic Number Theory, Science Press, Beijing, 99.

96 A CLASS OF DIRICHLET SERIES AND ITS IDENTITIES Lou Yuanbing College of Science, Tibet University, Lhasa, Tibet, P.R.China Abstract Keywords: In this paper, we using the elementary methods to study the convergent properties of one class Dirichlet series involving a special sequences, and give an interesting identity for it. Convergent property; Dirichlet series; Identity.. Introduction and results For any positive integer n and m, we define the m-th power complement b m n is the smallest positive integer such that nb m n is a complete m-th power, see problem 9 of []. Now for any positive integer k, we also define an arithmetic function δ k n as follows: δ k n = { max{d N d n, d, k = }, if n 0, 0, if n = 0. Let A denotes the set of all positive integers n such that the equation δ k n = b m n. That is, A = {n : n N, δ k n = b m n}. In this paper, we using the elementary methods to study the convergent properties of the Dirichlet series involving the set A, and give an interesting identity for it. That is, we shall prove the following conclusion: Theorem. Let m be a positive even number. Then for any real number s > and positive integer k, we have the identity: n= n A n s = ζ m s ζms p ms p k p 3 ms p ms where ζs is the Riemann zeta-function, and denotes the product over all p primes. From this Theorem we may immediately deduce the following corollaries:,

97 88 RESEARCH ON SMARANDACHE PROBLEMS IN NUMBER THEORY II Corollary. Let B = {n : n N, δ k n = b n}, then we have n= n B n = 5 π p k p 6 p 4 p. Corollary. Let C = {n : n N, δ k n = b 4 n}, then we have n= n C n = n= n B n 4 = 05 π 4 p k p p 8 p 4. Corollary 3. Let C = {n : n N, δ k n = b 4 n}, then we have n 3 = π 6 n= n C p k p 8 p p 6.. Proof of the theorem In this section, we will complete the proof of the theorem. For any real number s > 0, it is clear that and n= n= n A n s < n= n s is convergent if s >, thus n= n A n s, is also convergent if s >. ns Now we find the set A. Let n = p α pα pαs s denotes the factorization of n into prime powers. First from the definitions of δ k n and b m n we konw that δ k n and b m n both are multiplicative functions. So in order to find A, we only discuss the problem in case n = p α. If n = p α and p, k =, then we have δ k p α = p α ; b m p α = p m α, if α m; b m p α = p m+m[ α m ] α, if α > m and α m; b m p α =, if α = rm, where r is any positive integer, and [x] denotes the greatest integer x. So in this case δ k p α = b m p α if and only if α = m. If n = p α and p, k, then δ k p α =, so in this case the equation δ k p α = b m p α has solution if and only if n = p rm, r = 0,,,. Now from the Euler product formula see [] and the definition of A, we have n= n A n s = p k + p m s p k + p ms + p ms + + p3ms

98 A class of Dirichlet series and its identities 89 = p + p m s = ζ m s ζms p k p k p ms p ms p 3 ms + p m s p k, p ms + where ζs is the Riemann zeta-function, and denotes the product over all p primes. This completes the proof of Theorem. The Corollaries follows from ζ = π /6, ζ4 = π 4 /90, ζ6 = π 8 /945, ζ8 = π 8 /9450 and ζ = References 69π [] F.Smarandache. Only problems, not solutions, Xiquan Publishing House, Chicago, 993. [] Tom M.Apostol. Introduction to Analytic Number Theory. New York: Springer-Verlag, 976.

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100 AN ARITHMETICAL FUNCTION AND THE PERFECT K-TH POWER NUMBERS Yang Qianli and Yang Mingshun Department of Mathematics, Weinan Teacher s College, Weinan, Shaanxi, P.R.China Abstract Keywords: For any primes p and q with p, q =, the arithmetical function e pqn defined as the largest exponent of power pq which divides n. In this paper, we use the elementary methods to study the mean value properties of e pqn acting on the perfect k-th power number sequences, and give an interesting asymptotic formula for it. the perfect k-th power number; Asymptotic formula; Mean value.. Introduction For any prime p, let e p n denotes the largest exponent of power p which divides n. In problem 68 of reference [], Professor F.Smarandache asked us to study the properties of this arithmetical function. About this problem, many scholars showed great interest in it, and obtained some interesting results, see references [] and [3]. Similarly, we will define arithmetical function e pq n as follows: for any two primes p and q with p, q =, let e pq n denotes the largest exponent of power pq which divides n. That is, e pq n = max{α : pq α n, α N + }. According to [], a number n is called a perfect k-th power number if it satisfied k α for all p α n, where p α n denotes p α n, but p α+ n. Let A denotes the set of all the perfect k-th power numbers. It seems that no one knows the relations between these two arithmetical functions before. The main purpose of this paper is using the elementary methods to study the mean value properties of e pq n acting on the set A, and give an interesting asymptotic formula for it. That is, we shall prove the following: Theorem. Let p and q are two primes with p, q =, then for any real number x, we have the asymptotic formula n A e pq n = C p,q kx k + x k +ɛ,

101 9 RESEARCH ON SMARANDACHE PROBLEMS IN NUMBER THEORY II where C p,q = p q pq n= n pq n is a computable positive constant, and ɛ denotes any fixed positive number.. Proof of the theorem In this section, we shall complete the proof of the theorem. First we define arithmetical function an as follows: {, if n is a perfect k-th power number; an = 0, otherwise. In order to complete the proof of Theorem, we need the following: Lemma. For any real number x, we have the asymptotic formula Proof. Let n,pq= an = x k fs = p q pq n= n,pq= an n s. + O x +ɛ k. where Res >. From the Euler product formula [4] and the multiplicative properties of an, we have fs = + ap k P ks + ap k P ks + P P,pq= + = P = ζks p ks P ks + + P ks q ks, p ks q ks where ζs is the Riemann zeta-function, and P denotes the product over all primes. Now by Perron formula [5] with s 0 = 0, b = k + log x, T = x k, Hx = x and Bσ =, we have σ k n,pq= an = b+it πi b it ζks pks q ks pq ks x s ds + Ox k +ε. s

102 An arithmetical function and the perfect k-th power numbers 93 To estimate the main term, we move the integral line from b = k + log x to a = k + log x Therefore, b+it a+it a it b it πi b it b+it a+it a it [ fs xs s ds = Res fs xs s, ]. k Note that lim ζss =, we may immediately get s Res [fs xs s, ] = x k. k p q Now from the estimate a+it a it + + πi b+it a+it we can easily get n,pq= an = x k b it a it p q pq fs xs s ds x + O x +ɛ k. k +ɛ, This proves the lemma. Now we prove the theorem. From the properties of geometrical series and the definition of e pq n, combining the lemma we have = = e pq n n A α α log pq x n x pq α k α n,pq= = kx k = kx k α log pq x k kα an x pq kα p q pq p q pq +O x +ɛ k n= k p q x +ɛ k + O pq pq kα n= n pq n α> log pq x k n pq n pq [ logpq x k α pq α + O ] α + α= x +ɛ k [ logpq x k pq α ]

103 94 RESEARCH ON SMARANDACHE PROBLEMS IN NUMBER THEORY II = kx p q k a p,q + Ox k log x + O x +ɛ k pq p q = a p,q kx k + x +ɛ k, pq where a p,q = n= is a computable positive constant. This completes the proof of Theorem. References n pq n [] F.Smarandache. Only problems, not Solutions. Xiquan Publ.House, Chicago, 993. [] Lv Chuan. A number theoretic function and its mean value, Research on Smarandache problems in number theory collected papers. Hexis, 004, [3] Zhang Wenpeng. An arithmetic function and the primitive number of power p, Research on Smarandache problems in number theory collected papers. Hexis, 004, -4. [4] T.M.Apostol. Introduction to Analytic Number Theory. Springer-Verlag, 976. [5] Pan Chengdong and Pan Chengbiao. Elemens of the analytic number Theory. Beijing, Science Press. 99.

104 AN ARITHMETICAL FUNCTION AND ITS MEAN VALUE FORMULA Ren Zhibin and Zhao Xiaopeng Department of Mathematics, Weinan Teacher s College, Weinan, Shaanxi, P.R.China Abstract Keywords: In this paper, we use the elementary methods to study the mean value properties of an arithmetical function, and give an interesting asymptotic formula for it. Arithmetical function; Mean value; Asymptotic formula.. Introduction Let k be a fixed positive integer. For any positive integer n, we define the arithmetical function b k n as follows: { } m b k n = max m i k n, n N. i= m That is, b k n is the greatest positive integer m such that i k n. For i= example, b =, b =, b 3 =, b 4 =, b 5 =,. In fact, from the definition of b k n we know that if n < + k, then m bn = ; if + k n < + k + 3 k, then bn = ;, if i k i= m+ n < i k, then bn = m. So the same positive integer m repeated m+ k i= times in the sequence {b k n} n =,, 3, 4,. About this arithmetical function, it seems that none had studied it, at least we have not see any related references. In this paper, we shall use the elementary methods to study the mean value properties of b k n, and give an interesting asymptotic formula for it. That is, we shall prove the following: Theorem. For any real number x >, we have the asymptotic formula b k n = k+ k + k+ x k+ k+ + Ox. k + From the Theorem, we may immediately deduce the following two Corollaries

105 96 RESEARCH ON SMARANDACHE PROBLEMS IN NUMBER THEORY II Corollary. Corollary. For any real number x >, we have the asymptotic formula b n = 3 3 x 3 + Ox. For any real number x >, we have the asymptotic formula. Proof of the theorem b n = x Ox. In this section, we shall complete the proof of the theorem. First we define n f k n = i k. For any real number x >, it is clear that there exists one and i= only one positive integer N such that f k N x < f k N +. For any fixed positive integer k, from the Euler s summation formula see Theorem 3. of [] we have f k N = N i= i k = N k+ k + +O N k N + k+ = +O N k = f k N +. k + So that from the inequality f k N x < f k N + we have x = N k+ k + + O N k or k + k+ x k+ = N + O. Then from the Euler s summation formula and the definition of b k n, we have b k n = = N n= f k n j<f k n+ N n= b k j + n + k n + ON k+ = f k N n<x N n= = N k+ k + N k+ k + + ON k+ = N k+ = k+ k + k+ x k+ k+ + Ox. k + This completes the proof of the theorem. References b k n n k n + ON k+ k + + ON k+ [] F. Smarandache. Only Problems, Not solutions. Chicago: Xiquan Publishing House, 993.

106 An arithmetical function and its mean value formula 97 [] Tom M.Apostol. Introduction to Analytic Number Theory. New York: Springer-Verlag, 976. [3] Zhang Wenpeng. Research on Smarandache problems in number theory. Hexis Publishing House, 004.

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108 ON THE SQUARE COMPLEMENTS FUNCTION OF N! Fu Ruiqin School of Science, Xi an Shiyou University, Xi an, Shaanxi, P.R.China Yang Hai Research Center for Basic Science, Xi an Jiaotong University, Xi an, Shaanxi, P.R.China Abstract Keywords: For any positive integer n, let bn denote the square complements function of n. That is, bn denotes the smallest positive integer such that n bn is a perfect square number. In this paper, we use the elementary method to study the asymptotic properties of bn!, and give an interesting asymptotic formula for lnbn!. Square complements function; Standard factorization; Asymptotic formula.. Introduction For any positive integer n, let bn denote the square complements function of n. That is, bn denotes the smallest positive integer k such that nk is a perfect square. For example, b =, b =, b3 = 3, b4 =, b5 = 5, b6 = 6, b7 = 7, b8 =,. In problem 7 of [], Professor F. Smarandache ask us to study the properties of bn. About this problem, some authors had studied it before, and obtained some interesting results, see references [5] and [6]. In this paper, we use the elementary method to study the asymptotic properties of the square complements function bn! of n, and give an interesting asymptotic formula for lnbn!. That is, we shall prove the following: Theorem. For any positive integer n, we have the asymptotic formula ln bn! = n ln + O n exp A ln 3 5 n ln ln n 5, where A > 0 is a constant.

109 00 RESEARCH ON SMARANDACHE PROBLEMS IN NUMBER THEORY II. Two simple lemmas To complete the proof of the theorem, we need the following two simple lemmas: Lemma. Let n! = p α pα pα k k denotes the factorization of n into prime powers. Then we have the calculate formula bn! = bp α bpα bpα k k = p ordp p ordp p ordp k, where the ord function is defined as: {, if αi is odd, ordp i = 0, if α i is even. Proof. See reference []. Lemma. For any real number x, we have the asymptotic formula θx = ln p p x = x + O Proof. See references [3] or [4]. 3. Proof of the theorem x exp A ln 3 5 x ln ln x 5 In this section, we shall complete the proof of Theorem. First from Lemma we have. ln bn! = ln bp α bpα bpα k k = ln p = p n ordp ln p + n <p n n 4 <p n 3 ln p + n 6 <p n 5 ln p + + O. Let n be a positive integer large enough, if a prime factor p of n! in the interval n, n], then the power of p is in the standard factorization of n!. Similarly, if a prime factor p of n! in the interval n 3, n ], then the power of p is in the standard factorization of n!; If a prime factor p of n! in the interval n 4, n 3 ], then the power of p is 3 in the standard factorization of n!,. On the other hand, note that the identity = ln, 4

110 On the square complements function of n! 0 then from Lemma we have ln bn! = θ n θ n n n n n + θ θ + θ θ + + O = n A ln n + O n exp ln ln n 5 3 A ln = n ln + O n exp 5 n ln ln n 5 This completes the proof of Theorem. References. [] Smarandache F. Only problems, not Solutions. Chicago: Xiquan Publ. House, 993. [] Felice Russo. An introduction to the Smarandache square complements. Smarandache Notions Journal, 00, 3:60-7. [3] Pan Chengdong and Pan Chengbiao. Foundation of Analytic Number Theory. Beijing: Science Press, 997. [4] J.B.Rosser and L.Schoenfeld. Approximates formulas for some functions of prime numbers. Illinois J. Math. 96, 6: [5] Liu H.Y. and Gou S. On a problem of F. Smarandache, Journal of Yanan University Natural Science Edition 00, 0: 5-6. [6] Zhang H.L. and Wang Y. On the mean value of divisor function for square complements, Basic Science Journal of Textile University 00,5:

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112 ON THE SMARANDACHE FUNCTION Wang Yongxing Department of Mathematics, Weinan Teacher s college, Weinan, Shaanxi, P.R.China Abstract Keywords: For any positive integer n, let Sn denotes the Smarandache function. In this paper, we study the mean value properties of Sn, and give an interesting asymptotic formula for n it. Smarandache function; Mean value; Asymptotic formula.. Introduction For any positive integer n, let Sn denotes the Smarandache function. That is, Sn is the smallest positive integer m such that n m!. From the definition of Sn, we can easily deduce that if n = p α pα pα k k is the prime powers factorization of n, then Sn = max i k {fpα i i }. About the arithmetical properties of Sn, many scholars had studied it before see reference []. The main purpose of this paper is to study the mean value properties of Sn n, and obtain an interesting asymptotic formula for it. That is, we shall prove the following: Theorem. For any real number x, we have the asymptotic formula Sn n = π x x 6 ln x + O ln. x. Some lemmas To complete the proof of Theorem, we need the following several Lemmas: Lemma. For any positive integer n, if n has the prime powers factorization n = p α pα pαr r P n such that P n > n, then we have the identity Sn = P n. where P n denotes the greatest prime divisor of n. Proof. From the prime powers factorization of n, we may immediately get p α pα pαr r < n.

113 04 RESEARCH ON SMARANDACHE PROBLEMS IN NUMBER THEORY II Then we have p α i i P n! i =,, r. Thus we can easily obtain n P n!. But P n P n!, so we have Sn = P n. This completes the proof of Lemma. Lemma. For any real number x, we have the asymptotic formula Sn = π x x ln x + O ln. x Proof. First we define two sets A and B as following: and A = {n n x, P n n} B = {n n x, P n > n}. Using the Euler s summation formula see reference [], we may get n ln n Sn n A x x = t ln tdt + t [t] t ln t dt + x ln xx [x] x 3 ln x. Similarly, from the Abel s summation forumula we also have Sn = n B P n> n = P n n x n p x n = p n x x p x n = n x x n π x n p + O n x n p x n xπ x x x n x πsds + O x 3 ln x, where πx denotes all the numbers of primes which is not exceeding x. Note that πx = x x ln x + O ln. x

114 On the Smarandache function 05 Using the above asymptotic formula, we have x p x n xπ x n x πsds x p = x x n π n = x n ln x/n x ln x + O x +O ln + O x Considering the following n x n = ζ + O x. x n ln x/n x n ln x/n x ln x. Hence we have x n n ln x x n = = π 6 n ln x x n ln x + O n x x ln x + O ln x ln x n x x n ln x n and x n n ln x x n = O x ln x. Combining all the above, we may immediately deduce that Sn = Sn + Sn n A n B = π x x ln x + O ln. x This completes the proof of Lemma. 3. Proof of the theorem In this section, we shall complete the proof of Theorem. First applying the Abel s summation, and note that the results of Lemma and Lemma, we may have Sn n = x x Sn + t St dt n t

115 06 RESEARCH ON SMARANDACHE PROBLEMS IN NUMBER THEORY II = x = π = π 6 π x x ln x + O ln x x x ln x + O + ln x x x ln x + O ln x This completes the proof of Theorem. References. x π t t + t ln t + O ln dt t x π x ln t dt + O ln t dt [] Li Hailong and Zhao Xiaopeng. On the Smarandache function and the k-th roots of a positive integer. Hexis, 9-, 004. [] Tom M A. Introduction to Analytic Number Theory. New York: Springer- Verlag, 976.

116 MEAN VALUE OF THE K-POWER COMPLEMENT SEQUENCES Feng Zhiyu Fengxiang Teacher s School, Fengxiang, Shaanxi, P.R.China Abstract Keywords: For any prime p 3 and any fixed integer k, let e pn denote the largest exponent of power p which divides n, an, k denotes the k-power complement number of n. In this paper, we study the properties of the sequence e pan, k, and give an interesting asymptotic formula for it. Largest exponent; Mean value; Asymptotic formula.. Introduction Let p 3 be a prime, e p n denote the largest exponent of power p which divides n. For any fixed integer k, let an, k denote k-power complement sequence of n. That is, an, k is the smallest positive integer such that nan, k is a perfect k-power. In problem 9 of reference [], professor F. Smarandach asked us to study the properties of this sequences. About this problem, some people had studied it before, and made some progress, see reference []. The main purpose of this paper is using the analytic method to study the properties of the sequence e p an, k, and give an interesting asymptotic formula for its mean value. That is, we shall prove the following: Theorem. Let p 3 be a prime and k a fixed positive integer. Then for any real number x, we have the asymptotic formula e p an, k = k pk kp k + p k x + O x /+ε, p where ε denotes any fixed positive number. From this Theorem we may immediately deduce the following two corollaries: Corollary. Let p 3 be a prime. Then for any real number x, we have the asymptotic formula e p an, = p x + O x /+ε.

117 08 RESEARCH ON SMARANDACHE PROBLEMS IN NUMBER THEORY II Corollary. Let p 3 be a prime. Then for any real number x, we have e p an, 3 = p + p + p + x + O x /+ε.. Proof of the theorem In this section, we shall complete the proof of Theorem. In fact, for any complex number s with Res >, we define the Dirichlet series fs = n= e p an, k n s. From the definitions of e p n and an, k, and applying the Euler product formula See reference [4] we have = = k = = fs k α=0 β= k α=0 β= k β p αk+βs p αk+βs n = n,p= n = n,p= n s k p s p ks ζs p ks p s p k s [k p s k] p ks p s = k pks kp k s + p ks p s ζs, n s α=0 p s k p αks β= p ks ζs + k p ks ζs β p βs where ζs is the Riemann zeta-function. Obviously, we have e p an, k e p an, k k log p n k ln n n σ n= n = n,p= n s p s p ks k p p ks ζs s k σ, where σ is the real part of s. Therefore by Parron s formula See reference [3] we can get e p an, k n s = b+it fs + s 0 0 xs x b πi b it s ds + O Bb + σ 0 T +O { x σ 0 Hx min, ln x T } { + O x σ 0 HN min, } x x

118 Mean value of the k-power complement sequences 09 where N is the nearest integer to x, x = x N. Taking s 0 = 0, b = 3, Hx = k ln x, Bσ = k where e p an, k = πi To estimate the main term 3 +it σ, we have 3 +ε ζsrs xs ds + Ox 3 it s T, Rs = k pks kp k s + p ks p s. πi 3 +it ζsrs xs 3 it s ds, we move the integral line from s = 3 ± it to s = ± it. This time, the function gs = ζsrs xs s has a simple pole point at s =, and the residue is So we have 3 +it πi + 3 it = Note that the estimate +it πi x 3 +ε T it k p k kp k + p k x. p +it + 3 +it 3 it e p an, k = it + +it 3 it it k p k kp k + + it p k p it +it x. ζsrs xs s ds ζs k pks kp k s + x s p ks p s s ds Taking T = x, from the above formula we may immediately get the asymptotic formula k p k kp k + x p k p This completes the proof of Theorem. + O x /+ε.

119 0 RESEARCH ON SMARANDACHE PROBLEMS IN NUMBER THEORY II References [] F. Smarandache. Only Problems, Not solutions. Chicago: Xiquan Publishing House, 993. [] Zhang Wenpeng. Reseach on Smarandache problems in number theory. Hexis Publishing House, 004. [3] Pan Chengdong and Pan Chengbiao. Foundation of Analytic Number Theory. Beijing: Science press, 997. [4] Apostol T M. Introduction to Analytic Number Theory. New York: Springer- Verlag, 976.

120 ON THE M-TH POWER FREE NUMBER SEQUENCES Chen Guohui Department of Mathematics, Hainan Normal University, Haikou, Hainan, P.R.China Abstract Keywords: In this paper, we using the elementary methods to study the arithmetical properties of the m-th power free number sequences, and give some interesting identities for it. m-free number sequence; Dirichlet s series; Identity.. Introduction For any positive integer n and m with m, we define the m-th power free number function a m n of n as follows: If n = p α pα pαs s is the prime powers decomposition of n, then the m-th power free number function of n is the function: a m n = p β pβ pβs s, where β i = α i, if α i m, and β i = 0, if α i m. For any positive integer k, we also define the arithmetical function δ k n as follows: { max{d N d n, d, k = }, if n 0, δ k n = 0, if n = 0. Let A denotes the set of all the positive integers n such that the equation a m n = δ k n. That is, A = {n N, a m n = δ k n}. In this paper, we using the elementary methods to study the convergent properties of the Dirichlet series involving the set A, and give an interesting identity. That is, we shall prove the following conclusion: Theorem. Let m be a positive integer. Then for any real number s >, we have the identity: n= n A n s = ζs ζms p k where ζs is the Riemann zeta-function. p ms p m s + p ms, From this theorem we may immediately deduce the following:

121 RESEARCH ON SMARANDACHE PROBLEMS IN NUMBER THEORY II Corollary. Let B = {n N, a n = δ k n} and C = {n N, a 3 n = δ k n}, then we have the identities: and n= n C n= n B n = 5 π p k n = 305 π 4 p k p 4 p + p 4 p 6 p 4 + p 6.. Proof of the theorem In this section, we will complete the proof of the theorem. First, we define the arithmetical function an as follows: {, if n A, an = 0, if otherwise. and For any real number s > 0, it is clear that n= n= n A n s = n= an n s < n s is convergent if s >, thus n= n A n= n s, is also convergent if s >. ns Now let n = p α pα pαs s denotes the factorization n into prime powers. Then from the definition of a m n and δ k n, we know that if α j m for some j, then a m p α j j =. So only when α i m and p i, k = for all i, the equation a m n = δ k n has solution. If p j, k for some j, then the equation a m p α j j = δ kp α j j has solution if and only if α j m. So from the Euler product formula see [], we have n s = + ap p p s + ap p s + + apm p m s + n= n A = + ap p s + ap p s + + apm p p k m s + ap p ms + ap p m+s + ap3 p m+s + p k = + p s + p s + + p p k m s

122 On the m-th power free number sequences 3 p k = ζs ζms p k + p ms + p m+s + p m+s + p ms p m s + p ms, where ζs is the Riemann zeta-function and p primes. This completes the proof of Theorem. denotes the product over all References [] Tom M.Apostol. Introduction to Analytic Number Theory. New York: Springer-Verlag, 976. [] F.Smarandache. Only problems, not solutions, Xiquan Publishing House, Chicago, 993.

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124 A NUMBER THEORETIC FUNCTION AND ITS MEAN VALUE Du Jianli School of Science, Xi an Shiyou University, Xi an, Shaanxi, P.R.China Abstract Keywords: Let p and q are two distinct primes, e pqn denotes the largest exponent of power n pq which divides n, and let b pqn = e pq e pqt. In this paper, we t using the analytic methods to study the mean vlaue properties of the b pqn, and give an interesting asymptotic formula for it. t n Largest exponent; Mean value; Asymptotic formula.. Introduction Let p and q are two distinct primes, e pq n denotes the largest exponent of power pq which divides n. In problem 68 of [3], professor F.Smarandache asked us to study the properties of the sequence e p n. About this problem, some people had studied it, and obtained a series interesting results, see references []. In this paper, we define arithmetical function b pq n = n e pq e pq t, then we using the analytic methods to study the mean t t n value properties of b pq n, and give give a sharp asymptotic formula for it. That is, we shall prove the following: Theorem. Let p and q are two distinct primes, then for any real number x, we have the asymptotic formula b pq n = x ln x pq γ pq + pqγ pq lnpq + pq 3 where ε is any fixed positive number, and γ is the Euler constant.. Proof of the theorem x + O x +ε, In this section, we shall complete the proof of Theorem. For any complex s, we define the Dirichlet s series fs = n= e pq n n s and gs = n= b pq n n s.

125 6 RESEARCH ON SMARANDACHE PROBLEMS IN NUMBER THEORY II It is clear that gs = f s. Ren Ganglian has given in a paper to appear fs = ζs pq s, where ζs is the Riemann zeta-function. So we can obtain that gs = ζ s. pq s Obviously, we have b pq n b pq n log pq n n ln n n σ σ, where σ is the real part of s. Therefore, by Parron s formula See reference [4] we can get b pq n n s = b+it ζ s + s 0 0 Rs + s 0 xs x b πi s ds + O Bb + σ 0 T +O b it { x σ 0 Hx min, log x T n= } { +O x σ 0 HN min, where N be the nearest integer to x, x = x N. Let Rs = pq s. Taking s 0 = 0, b = 5, Hx = x ln x, Bσ = σ, we have b pq n = πi To estimate the main term πi 5 +it 3 +ε ζ srs xs ds + Ox 5 it s T. 5 +it ζ srs xs 5 it s ds, } x, x we move the integral line from s = 5 ± it to s = ± it. This time, the function ζ srs xs s has a second order pole point at s =, and the residue is where γ is Euler constant. So we have 5 +it πi x ln x γ pq + pqγ pq lnpq + pq pq 3 x, + 5 it +it + 5 +it it + +it 5 it it ζ srs xs s ds = x ln x γ pq + pqγ pq lnpq + pq pq 3 x.

126 A number theoretic function and its mean value 7 and Taking T = x, we can easily obtain +it 5 it ζ srs xs πi s ds x +ε + 5 +it πi it it ζ srs xs +it s ds x +ε. So from the above formula, we may immediately get the asymptotic formula b pq n = x ln x pq This completes the proof of Theorem. References γ pq + pqγ pq lnpq + pq 3 x+o x /+ε. [] Apostol T M. Introduction to Analytic Number Theory. New York: Springer- Verlag, 976. [] Zhang Wenpeng. Reseach on Smarandache problems in number theory. Hexis Publishing House, 004. [3] F. Smarandache. Only Problems, Not solutions. Chicago: Xiquan Publishing House, 993. [4] Pan Chengdong and Pan Chengbiao. Foundation of Analytic Number Theory. Beijing: Science press, 997.

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128 ON THE ADDITIVE K-TH POWER PART RESIDUE FUNCTION Zhang Shengsheng Shaanxi Technical College of Finance and Economics, Xianyang, Shaanxi, P.R.China Abstract Similar to the Smarandache k-th power complements, we define the additive k- th power part residue f k n of n is the smallest nonnegative integer such that n f k n is a perfect k-th power. The main purpose of this paper is using the elementary methods to study the mean value properties of f k n and df k n, and give two interesting asymptotic formulae for them. Keywords: Additive k-th power part residue function; Mean value; Asymptotic formula.. Introduction and results For any positive integer n, the Smarandache k-th power complements b k n is the smallest positive integer b k n such that nb k n is a complete k-th power see problem 9 of []. Similar to the Smarandache k-th power complements, Xu Zhefeng in [] defined the additive k-th power complements a k n as follows: a k n is the smallest nonnegative integer such that n + a k n is a complete k-th power. Similarly, we will define the additive k-th power part residue f k n as following: for any positive integer n, f k n = min{r 0 r = n m k, m N}. For example, if k =, we have the additive square part residue sequences {f n} n =,,... as following: 0,,, 0,,, 3, 4, 0,,, 3, 4, 5, 6, 0,,,.... In this paper, we use the elementary methods to study the mean value properties of f k n and df k n where dn is the Dirichlet divisor function, and obtain some interesting asymptotic formulae for them. That is, we will prove the following conclusions: Theorem. For any real number x 3 and integer k, we have the asymptotic formula f k n = k k x k + O x k.

129 0 RESEARCH ON SMARANDACHE PROBLEMS IN NUMBER THEORY II Theorem. For any real number x 3 and integer k, we also have the asymptotic formula df k n = x ln x + γ + ln k + x + O x k ln x, k k where γ is the Euler constant.. Some Lemmas To complete the proof of the theorems, we need following two Lemmas. First we have Lemma. For any real number x 3, we have the asymptotic formula dn = x ln x + γ x + Ox, where γ is the Euler constant. Proof. See reference [3]. Lemma. For any real number x 3 and any nonnegative arithmetical function hn with h0 = 0, we have the asymptotic formula M hf k n = hn + O hn, where gt = k i= t= n gt x gm k t i and M = [x k ], [x] denotes the greatest integer not i exceeding x. Proof. For any real number x, let M be a fixed positive integer such that M k x < M + k. Noting that if n pass through the integers in the interval [t k, t + k, then f k n pass through the integers in the interval [0, t+ k t k ] and h0 = 0, we can deduce that hf k n = = = = M t= M t= M t= M t= t k n<t+ k hf k n + n gt n gt n gt hn + M k hn 0 n<x M k hf k n hn + O hn 0 n M+ k M k hn + O hn, n gm

130 On the additive k-th power part residue function k k where gt = t i and M = [x k ]. i i= This completes the proof of Lemma. 3. Proof of the theorems In this section, we shall use the elementary method to complete the proof of the theorems. First we prove Theorem. Let hn = n and M = [x k ], note that x k M = O, then from Lemma and the Euler summation formula see [4] we obtain f k n = = = M t= M t= n gt n + O n gm k t k + Ox k n k k x k + Ox k. This completes the proof of Theorem. Now we prove Theorem. From Lemma and Lemma we have = = = df k n M t= M t= n gt dn + O n gm dn [ kt k ln kt k ] + γ kt k + O kt k +O x k ln x M t= kk t k ln t + γ + ln k kt k + Ot k +O x k ln x M M = kk t k ln t + γ + ln k k +O t= x k ln x. t= t k

131 RESEARCH ON SMARANDACHE PROBLEMS IN NUMBER THEORY II Then from the Euler summation formula, we can easily get x ln x + γ + ln k + k k df k n = This completes the proof of the theorems. References x + O x k ln x. [] F.Smarandache. Only Problems, not Solutions. Xiquan Publishing House, Chicago, 993. [] Xu Zhefeng. On the Additive k-power Complements. Research on Smarandache Problems in Number Theory, Hexis 3-6, 004. [3] G.L.Dirichlet. Sur I usage des Series infinies dans la Theorie des nombres. Crelle s Journal, 938 [4] Tom M. Apostol. Introduction to Analytic Number theory. New York: Springer-Verlag, 976. [5] Pan Chengdong and Pan Chengbiao. Foundation of Analytic Number Theory. Beijing: Science Press, 997, P98.

132 AN ARITHMETICAL FUNCTION AND THE K-TH POWER COMPLEMENTS Huang Wei Department of Engineering Course, Baoji Vocational Technology College, Baoji, Shaanxi, P.R.China Abstract Keywords: The main purpose of this paper is using the analytic methods to study the asymptotic properties of an arithmetical function acting on the k-th power complements, and give an interesting asymptotic formula for it. Arithmetical function; Asymptotic formula; k-th power complements.. Introduction and results For any positive integer n, let Dn denotes the number of the solutions of the equation n = n n with n, n =. That is, Dn = d n d, n d =. It is clear that Dn is a multiplicative function, and has many interesting arithmetical properties. Now we define another arithmetical function A k n. For any fixed positive integer k, let A k n denotes the k-th power complements. That is, A k n denotes the smallest positive integer such that na k n is a perfect k-th power. For example, A =, A =, A 3 = 3, A 4 =, A 5 = 5, A 6 = 6, A 7 = 7, A 8 =,. In reference [], professor F.Smarandache asked us to study the properties of the sequences {A k n}. About this problem, some people had studied it before, and obtained some interesting results, see references [4] and [5]. In this paper, we use the analytic methods to study the mean value properties of the arithmetical function Dn acting on the set {A k n}, and obtain a sharper asymptotic formula. That is, we shall prove the following: Theorem. For any real number x, we have the asymptotic formula DA k n = 6ζkx ln x π p p k + p k + Ckx + Ox +ɛ, where Ck is a computable constant, ζk is the Riemann zeta-function, ɛ denotes any fixed positive number, and denotes the product over all primes. p

133 4 RESEARCH ON SMARANDACHE PROBLEMS IN NUMBER THEORY II From this theorem we may immediately deduce the following: Corollary. For any real number x, we have the asymptotic formula. A Lemma DA k n = π 5 x ln x p p 4 + p 3 + C4x + Ox +ɛ. Before the proof of the theorem, a Lemma will be useful. Lemma. For any positive integer number n, we have the identity Dn = vn, where vn denotes the number of all distinct prime divisors of n, i.e. { 0, if n = ; vn = k, if n = p α pα pα 3 3 pα k k. Proof. Let n = p α pα pα 3 3 pα k k denotes the factorization of n into prime powers. Note that Dn is a multiplicative function and Dp α =, so from the definition of vn, we have Dn = d n d, n d = = Ck 0 + Ck + + Ck k = vn. This proves the lemma. 3. Proof of the theorem In this section, we shall complete the proof of the theorem. Let Dirichlet s series DA k n fs = n s. n= For any real number s >, it is clear that fs is absolutely convergent. So from the lemma and the Euler s product formula [] we have fs = p = p = p + DA kp p s + DA kp p s + + Dpk p s + Dpk p s + + D p ks + vpk vpk + p s + p s + + v p ks +

134 An arithmetical function and the k-th power complements 5 = p + p s + p s + + p ks + = ζsζks p = ζ sζks ζs p + p s p ks p ks + p k s p k+s + where ζs is the Riemann zeta-function. Therefore by Perron s formula [3] with s 0 = 0, b =, T = 3, we have where DA k n = πi To estimate the main term Rs = p πi +it it +it it, ζ sζks Rs xs ζs s ds + Ox +ε, p ks + p k s. ζ sζks Rs xs ζs s ds, we move the integral line from s = ± it to s = ± it, then the function ζk ζ x ln x p ζ sζks Rs xs ζs s have a second order pole point at s = with residue p k + p k where Ck is a computable constant. So we have +it + +it +it + πi = it 6ζkx ln x π p +it We can easily get the estimate +it +it + πi +it + it it + it it p k + p k it it + Ckx. + Ckx, ζsζksrs xs s ds ζsζksrs xs s ds x +ε. Now note that ζ = π /6, we may immediately obtain the asymptotic formula DA k n = 6 π ζkx ln x p p k + p k + Ckx + Ox +ɛ,

135 6 RESEARCH ON SMARANDACHE PROBLEMS IN NUMBER THEORY II where Ck is a computable constant. This completes the proof of the theorem. References [] F. Smarandache. Only Problems, Not solutions. Chicago: Xiquan Publishing House, 993. [] Apostol T M. Introduction to Analytic Number Theory. New York: Springer- Verlag, 976. [3] Pan Chengdong and Pan Chengbiao. Foundation of Analytic Number Theory. Beijing: Science press, 997. [4] Zhu Weiyi. On the k-power complement and k-power free number sequence. Smarandache Notions Journal, 004, 4: [5] Liu Hongyan and Lou Yuanbing. A Note on the 9-th Smarandache s problem. Smarandache Notions Journal, 004, 4:

136 ON THE TRIANGLE NUMBER PART RESIDUE OF A POSITIVE INTEGER Ji Yongqiang Danfeng Teacher s School, Shangluo, Shaanxi, P.R.China Abstract Keywords: For any positive integer n, let an denotes the triangle number part residue of n. That is, an = n kk+, where k is the greatest positive integer such that kk+ n. In this paper, we using the elementary methods to study the mean value properties of the sequences {an}, and give two interesting asymptotic formulae for it. The triangle number part residue; Mean value; Asymptotic formula.. Introduction and results For any positive integer n, let an denotes the triangle number part residue of n. That is, an = n kk+, where k is the greatest positive integer such that kk+ n. For example, a = 0, a =, a3 = 0, a4 =, a5 =, a6 = 0,. In reference [], American-Romanian number theorist Professor F. Smarandache asked us to study the arithmetical properties of this sequences. About this problem, it seems that none had studied it, at least we have not seen any related papers before. In this paper, we using the elementary methods to study the mean value properties of this sequences, and give two interesting asymptotic formulae for it. That is, we shall prove the following: Theorem. For any real number x 3, we have the asymptotic formula an = 3 x 3 + Ox. Theorem. formula For any real number x 3, we also have the asymptotic dan = x ln x + γ + ln 3 x + Ox 3, where dn is the Dirichlet divisor function provide d0 = 0, and γ is the Euler constant.

137 8 RESEARCH ON SMARANDACHE PROBLEMS IN NUMBER THEORY II Note: It is clear that if there exists a mean value formula for any arithmetical function fn, then using our methods we can also obtain an asymptotic formula for fan.. Proof of the theorems In this section, we shall complete the proof of the theorems. First we prove Theorem. For any real number x 3, let M be a fixed positive integer such that MM + x < Then from the definition of an, we have an = = = = M k= M k= M kk+ n< k+k+ k+k+ i kk+ k k= i=0 M k= = 6 M 3 + O i + O M M + M +. an kk + + O M On the other hand, note that the estimates Combing and, we have x<n< M+M+ i + O an M+M+ s MM+ s M. M x M < 3 M +. an = 3 x 3 + Ox. This completes the proof of Theorem. Now we prove Theorem. Using the similar method of proving Theorem, we have M k dan = di + k= i=0 MM+ < dan.

138 On the triangle number part residue of a positive integer 9 Note that the asymptotic formula see reference [] where γ is the Euler constant. We have dan = M k= dn = x ln x + γ x + O k ln k + γ k + O k 3 + O x 3, i x MM+ di = M ln M 4 M + γ M + OM Now combining and 3 we may immediately get dan = x ln x + γ + ln 3 x + Ox 3. This completes the proof of Theorem. References [] F.Smarandache. Only Problems, not Solutions. Chicago: Xiquan Publ. House, 993. [] Pan Chengdong and Pan Chengbiao. Foundation of Analytic Number Theory. Beijing: Science Press, 999. [3] Tom M.Apostol. Introduction to Analytic Number Theory. Springer- Verlag: New York, 976.

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140 AN ARITHMETICAL FUNCTION AND THE K-FULL NUMBER SEQUENCES Zhao Jiantang Department of Mathematics, Xianyang Teacher s College, Xianyang, Shaanxi, P.R.China Abstract Keywords: The main purpose of this paper is using the elementary methods to study the mean value properties of an arithmetical function acting on the k-full number sequences, and give an interesting asymptotic formula for it. k-full number sequence; Asymptotic formula; Mean value.. Introduction For any prime p, let e p n denotes the largest exponent of power p which divides n. In problem 68 of reference [], Professor F.Smarandache asked us to study the properties of this arithmetical function. About this problem, many scholars showed great interest in it and obtained some interesting results. For example, Lv Chuan [] used the elementary methods to studied the asymptotic properties of the mean value em p n, and gave the following asymptotic formula: e m p n = p p Ax + Ologm+ x, where A is a computable constant. Professor Zhang [3] studied the mean value of e p S p n, where S p n denotes the smallest integer m such that p n m!, and obtained e p S p n = p + p x + Oln3 x. In this paper, we shall use the elementary methods to study the mean value properties of e p n acting on the k-full number sequences, and give an interesting asymptotic formula for it. For convenience, first we give the definition of the k-full number. In fact, a number n is called a k-full number if p n p k n for any prime divisor p of n. Let A denotes the set of all the k-full numbers. It seems that no one had studied the relations between the arithmetical function e p n and the k-full numbers. In this paper, we shall prove the following:

141 3 RESEARCH ON SMARANDACHE PROBLEMS IN NUMBER THEORY II Theorem. Let p be a prime, then for any real number x, we have the asymptotic formula e p n = Cp, ka p kx k + x +ε k, where n A Cp, k = 6k π +, p p k k + q q + q k a p k is a computable positive constant, ε > 0 is any fixed real number, and denotes the product over all prime q. q. Proof of the theorem In this section, we shall complete the proof of Theorem. Let an denotes the character function of k-full number. That is, {, if n is a k-full number; an = 0, otherwise. It is clear that Let Dirichlet series n A n,p= fs = = an. n,p= n= n,p= an n s. It is clear that this series is convergent if Re s >. From the Euler product formula [4] and the definition of an, we have fs = + aqk q ks + aqk+ q q p k+s + = + q ks q p q s = ζks ζks p ks p k s + where ζs is the Riemann zeta-function. q + q ks + q s,

142 An arithmetical function and the k-full number sequences 33 Obviously, we have an, n= an n σ σ, k where σ is the real part of s. Therefore, by the Perron s formula [4] we have n,p= an = b+it πi b it ζks ζs Rsxs s ds + Ox k +ε, + q ks + q s. where Rs = p ks p k s + q Now moving the integral line from b to a = k, we have b+it a+it a it b it πi b it b+it a+it a it and fs xs s ds = Res [ fs xs s, k Note that lim ζks s s k =, we may immediately get k ]. Res [fs xs s, ] = kx k k ζ R. 3 k Combining,, 3 and the following estimates a it fs xs πi s ds x we can easily get where Cp, k = 6k π a+it πi b+it n,p= a+it b it + fs xs a it k +ɛ s ds x k +ɛ an = Cp, kx k + O x k +ɛ, +. p p k k + q q + q k Based on the definition of e p n and the above estimate, we have e p n = α an n A k α log p x n x p α n,p=

143 34 RESEARCH ON SMARANDACHE PROBLEMS IN NUMBER THEORY II = = Cp, kx k = Cp, kx k = Cp, kx k = Cp, kx k α x k x +ɛ k Cp, k + O p α p α k α log p x α + O x k α log p x p α k +ɛ α k k α log p x p α k n α α + O n= p n k α>log p x p α k α<k p α k n α + log p x n= p n k p k [log p x] α= p α k α<k p α k x +ɛ k a p k + Ox k log x + O = Cp, ka p kx k + x k +ɛ, where Cp, k = 6k π x k +ɛ α + O +, p p k k + q q + q k x k +ɛ a p k = n n= p n k is a computable positive constant. This completes the proof of Theorem. α<k α p α k References [] F.Smarandache. Only problems, not Solutions. Xiquan Publ.House, Chicago, 993. [] Lv Chuan. A number theoretic function and its mean value, Research on Smarandache problems in number theory collected papers. Hexis, 004, [3] Zhang Wenpeng. An arithmetic function and the primitive number of power p, Research on Smarandache problems in number theory collected papers. Hexis, 004, -4. [4] T.M.Apostol. Introduction to Analytic Number Theory. Springer-Verlag, 976.

144 ON THE HYBRID MEAN VALUE OF SOME SPECIAL SEQUENCES Li Yansheng and Gao Li College of Mathematics and Computer Science, Yan an University, Yan an, Shaanxi, P.R.China Abstract Keywords: The main purpose of this paper is to define a new arithmetic function by the m-th power complement numbers and the k-th power free numbers, and use the analytic methods to obtain some interesting asymptotic formulae for them. m-th power complement numbers; k-th power free numbers; Asymptotic formula; Arithmetic function.. Introduction and main results Let m be a fixed positive integer with m. For any positive integer n, we define the m-th power complement numbers a m n of n as the smallest positive integer such that na m n is a perfect m-th power. We also call a positive integer n as k-th power free number if it can not be divided by any p m, where p be a prime. We denotes the k-th power free number by c k n. It is clear that c k n = n d k n µd, where µd is the Möbius function. In reference [], professor F.Smarandache asked us to study the properties of the m-th power complement numbers and the k-th power free number sequences. About these sequences, some people had studied them, and obtained many interesting results, see references [], [3], [4] and [5]. In this paper, we introduce a new sequences fn = a m nc k n, then we use the analytic method to study the mean value properties of this sequences, and obtain some sharp asymptotic formulae. That is, we shall prove the following: Theorem. For any real numberx, we have the asymptotic formula a m nc k n = 6xm+ m + π Rm + + Oxm+ +ε, This work is supported by the P.N.S.F00JK30

145 36 RESEARCH ON SMARANDACHE PROBLEMS IN NUMBER THEORY II where ε denotes any fixed positive number, and Rm + = p k m+ + p p + p k m+ p k m+, if m k; Rm + = p + p m + p mm+ i j= p + p mm+ p m p jm+m+ p jm+m+ + k j=im+ p im+m+ p jm+ p +, if m < k. Theorem. For any real numberx, we have the asymptotic formula φa m nc k n = 6xm+ m + π R m + + Ox m+ +ε, where φn is the Euler function, ε denotes any fixed positive number, and R m + = + ppk m+ p k m+ p pp + p k m+ p k m+, If m k; R m+ = p p m + p mm+ Hm + p + p mm+ p m + Gm +, p + if m < k; i p jm+m+ p jm+m+ Hm + = j= p jm+m+ p and Gm + = k j=im+ p im+m+ p im+m p jm+. Taking k = in Theorem and Theorem, we may immediately deduce the following two Corollaries:

146 On the hybrid mean value of some special sequences 37 Corollary. For any real number x and integer m, we have the asymptotic formula a m nc n = 6 m + π xm+ + Ox m+ +ε. Corollary. For any real numberx and integer m, we have the asymptotic formula φa m nc n = 6xm+ m + π R m + + Ox m+ +ε, where R m + = p. pp +. Proof of the theorem In this section, we shall complete the proof of the theorems. First we prove Theorem. Let fs = n= a m nc k n n s, It is clear that a m n and c k n are multiplicative functions of n, so a m nc k n is also a multiplicative function of n. If the real part of s is large enough, then the Dirichlet s series fs is absolutely convergent. So for m k, from the Eurler s product formula [6] we have fs = p = p = + a mpc k p p s + a mp c k p p s + + a mp k c k p k p k s + p s m + + p s + p s + + p k 3s ζs m ζs m + p p k s p s m + p k s p k s ;

147 38 RESEARCH ON SMARANDACHE PROBLEMS IN NUMBER THEORY II if m k im k < i + m, i >, then fs = p + a mpc k p p s + + a mp m c k p m p ms + ampm+ c k p m+ p m+s + + a m p m c k p m p ms + + a mp im+ c k p im+ p im+s + + a mp k c k p k p k s = + p p s m + p s m + p s + + p m s + p m+s m + p s + + p m s k p i+m p i m+s im j=im+ p js p m s + p ms i ζs m = ζs m + j= p p s m + p ms p m s + k p im+s p js j=im+ p s m +. p jm+s p jm+s Obviously, we have the inequality a m nc k n n, n= a m nc k n n σ < σ m, where σ > m + is the real part of s. So by Perron formula [7] we have: an n s 0 = b+it fs + s 0 xs x b πi b it s ds + O Bb + σ 0 T +O x σ 0 Hxmin, log x T +O x σ 0 x HNmin,. T x

148 On the hybrid mean value of some special sequences 39 when N is the nearest integer to x, N = x, x = x N. Taking s 0 = 0, b = m +, T = x 3, Hx = x, Bσ = where a m nc k n = m++it iπ m+ it Rs =, we have: σ m ζs m ζs m Rsxs s ds + Oxm+ +ε, p k s + p p s m + p k s p k s, m k; p m s + p ms i p jm+s p + j= jm+s p p s m + p ms p m s + k p im+s p js j=im+ p s m + To estimate the main term iπ m++it m+ it, m < k. ζs m ζs m Rsxs s ds + Oxm+ +ε, we move the integral line from s = m + ± it to s = ± it. This time, the function ζs mxs fs = ζs ms Rs have a simple pole point at s = m + with residue So we have m++it iπ = m+ it m+ + +it m+ + it + m++it m+ +it x m+ Rm +. m + ζ m+ it m+ it x m+ Rm +. m + ζ ζs mx s ζs ms Rsds Taking T = x 3/, we can easy get the estimate m+ +it m+ it ζs mx s + πi m++it m+ it ζs ms Rsds m+ ζσ m + it ζσ m + it Rsx xm+ dσ = x m+ T T m+

149 40 RESEARCH ON SMARANDACHE PROBLEMS IN NUMBER THEORY II and πi T 0 ζs ms Rsds dt xm+ +ε. m+ it m+ +it ζs mx s ζ/ + it x m+ ζ + it t Note that ζ = π, from the above estimates we have 6 a m nc k n = 6xm+ m + π Rm + + Oxm+ +ε. This completes the proof of Theorem. Now we prove Theorem. Let f s = n= φa m nc k n n s, Then from Euler product formula [6] and the definition of φn, we also have if m k, then f s = p = p = p = + ϕpm p p s + ϕpm p p s + + ϕpm k p k p k s + pm p m p s + pm p m p s + + pm p m p k s + p s m + p s m + p s + + p k 3s + p s m+ p + + s ζs m ζs m p p k s + ppk s p k s pp s m + p k s p k s ;

150 On the hybrid mean value of some special sequences 3 4 if m < k im k < i + m, i >, then f s + pm p m p s + + pm p m p ms and where = p = p = + pm p m p m+s + + pm p m p ms + + pi+m p i+m + + pi+m p i+m p im+s + p s m+ + + p s m + p s m + p + + s p m+s m p m+s m p s + + p m s + p s + + p m s + p + + s p i m+s im p i m+s im+ p i+m p i+m p js + k j=im+ ζs m ζs m p i p k s p m s p m s + p s p s + + p m s p m s + pm s + p ms Hs p s m + p ms p m s Hs = pjm+s+ p jm+s p jm+s p Gs = j= k j=im+ p im+s p im+s p js. + Gs p s m, + By Perron formula [7] and the method of proving Theorem, we also have where φa m nc k n = m++it iπ m+ it R s = ζs m ζs m R s xs s ds+oxm+ +ε, + ppk s p k s p pp s m + p k s p k s, m k; + pm s + p ms Hs p p s m + p ms p m s + Gs p s m, m < k. + This completes the proof of Theorem.

151 46 RESEARCH ON SMARANDACHE PROBLEMS IN NUMBER THEORY II = ζs p ks ζks p q p s p m p β q p m β<k Obviously, we have inequality δ m fan n, p β q p m β k + p p s + p p s + + pk p k s + p s + + p βs + pβ p β k β s p β+s p βs n= δ m fan n σ < σ, where σ > is the real part of s. So by Perron formula? we have an n s 0 n A = b+it fs + s 0 xs x b iπ b it s ds + O Bb + σ 0 T +O x σ 0 Hx min, log x T +O x σ 0 x HN min, x, where N is the nearest integer to x, x = x N. Taking s 0 = 0, b =, T = x 3, Hx = x, Bσ = σ, we have δ m fn = +it ζs iπ it ζks Rsxs s ds + Ox +ε, where Rs = p ks p q p s p m p β q p m β<k To estimate the main term p β q p m β k + p p s + p p s + + pk p k s + p s + + p βs + pβ p β k β s p β+s p βs iπ +it it ζs ζks Rsxs s ds,..

152 4 RESEARCH ON SMARANDACHE PROBLEMS IN NUMBER THEORY II References [] F. Smarandache. Only problems, Not solutions. Chicago: Xiquan Publ. House, 993, P7. [] Zhang Tianping. On the m-th power free number sequence. Smarandache Notions Journal, Vol.4004, [3] Zhu Weiyi. On the k-th power complement and k-th power free number sequences. Smarandache Notions Journal, Vol.4004, [4] Zhang Tianping. On the cubic residues numbers and k-th power complement numbers. Smarandache Notions Journal, Vol.4004, [5] Yao Weili. On the mean value of the m-th power free numbers. Smarandache Notions Journal, Vol.4004, [6] Tom M. Apstol. Introduction to Analytic Number Theory. New York: Springer-Verlag, 976. [7] Pan Chengdong and Pan Chengbiao. Foundation of Analytic Number Theory. Beijing: Science Press, 997, P98.

153 ON THE MEAN VALUE OF A NEW ARITHMETICAL FUNCTION Ma Junqing Department of Mathematics, Xianyang Teacher s College Xianyang, Shaanxi, P.R.China Abstract Keywords: The main purpose of this paper is using the analytic method to study the mean value properties of δ mfn, and obtain an interesting asymptotic formula for it. Arithmetic function; k-power free numbers; Mean value.. Introduction and main results For any fixed positive integer q and any positive integer n, we define the arithmetical function fn = fq, n = q, n, where q, n denote the greatest common divisor of q and n. Obviously it is a multiplicative arithmetical function. Another famous multiplicative function is defined as δ t n = max{d : d n, d, t = }, where t is any fixed positive integer. In reference [], Professor J.Herzog and T.Maxsein studied the mean value of the error term E t n = δ t n tx σt, where σt = d t d, and proved that E t n = tx 4σt + O x ln ωt x, where ωt denotes the number of all different prime divisors of n. In this paper, we want to study the mean value properties of δ m fn, and obtain an interesting asymptotic formula for it. First we need to introduce a special number: k-power free number. A positive integer n is called a k-power free number if it can not be divided by any p k, where p is a prime number.

154 44 RESEARCH ON SMARANDACHE PROBLEMS IN NUMBER THEORY II In reference [], Professor F. Smarandache asked us to study the properties of the k-power free number sequence. About this problem, many scholars had studied it before see reference [3] and [4]. Furthermore, there exists an interesting identity δ t n = δ αt n, where αt is a square free number. In this paper, we shall use the analytic method to prove the following conclusion. Theorem. Let A denotes the set of all k-power free numbers k >. Then for any fixed positive integer m and any real number x, we have the asymptotic formula n A δ m fn = R ζk x + O x +ε, where ζk is the Riemann-zeta function, ε denotes any fixed positive number, and Rs = p q p m p ks p s p β q p m β k p β q p m β<k + p s + + p βs + pβ p β k β s p β+s p βs + p p s + p p s + + pk p k s where p β q denotes p β q and p β+ q. From this theorem we may immediately deduce the following three Corollaries: Corollary Let q be a square free number, then for any fixed positive integer m and any real number x, we have δ m fn = n A x ζk p q p m p k p k p k, + p k p + O x +ε. Corollary Let m be any fixed positive integer with q, m =. If q be a k-power free number, then we have δ m fn = n A x ζk p q p k p k p k β + + pβ+ k + O x +ε. p p β q

155 On the mean value of a new arithmetical function 45 Corollary 3 For any fixed positive integer q and any real number x, we have δ q fn = x ζk + O x +ε. n A. Proof of the theorem In this section, we shall complete the proof of Theorem. For convenience, we define a new arithmetical function an as follows: { n, ifn=, ornisak freenumber, an = 0, otherwise. It is clear that Let n = an. n A F s = n= δ m fan n s. From the Euler product formula [5] and the definition of δ m fan we have F s = + δ mfp p p s + δ mfp p s + + δ mfp k p k s = + δ mq, p p p s + δ mq, p p s + + δ mq, p k p k s = + p s + + p p q k s + p s + + p k s p m,q + δ mm, p p s + + δ mm, p k p k s p q p m = ζs ζks p q p m p β q p m β<k p ks p s p β q p m β k + p p s + + pβ p βs + + p p s + p p s + + pβ p β+s + + pk p k s pβ p k s

156 On the mean value of a new arithmetical function 47 we move the integral line from s = ± it to s = ± it. This time, the function fs = ζsxs ζkss Rs has a simple pole point at s = with residue R ζk x. So we have ζsx s +it + +it + iπ it +it it + +it it it We can easily get the estimate +it it ζsx s + πi +it it ζkss ds ζσ + it x ζkσ + it T and πi it ζsxs +it ζkss ds Combining the above estimates we have n A This completes the proof of Theorem. References Rx ds = ζkss ζk. dσ x T = x T ζ/ + it x 0 ζk/ + kit t + dt x +ε. δ m fn = R ζk x + O x +ε. [] J.Herzog and T.Maxsein. On the behavior of a certain error-term, Arch. Math., vol. 50, 988, [] F. Smarandache. Only problems, not Solutions, Xiquan Publ. House, Chicago, 993, 4-5. [3] Zhang Tianping. On the k-power free number sequence, Smarandache Notions Journal, vol. 4, 004, [4] Zhu Weiyi. On the k-power complement and k-power free number sequence, Smarandache Notions Journal, vol. 4, 004, [5] Tom M. Apostol, Introduction to Analytic Number Theory, Springer- Verlag, New York, 976. [6] Pan Chengdong and Pan Chengbiao. Foundation of Analytic Number Theory, Science Press, Beijing, 997.

157 This book contains 34 papers, most of which were written by participants to the First Northwest Number Theory Conference held in Shangluo Teacher s College, China, in March, 005. In this Conference, several professors gave a talk on Smarandache Problems and many participants lectured on them both extensively and intensively. All these papers are original and have been refereed. The themes of these papers range from the mean value or hybrid mean value of Smarandache type functions, the mean value of some famous number theoretic functions acting on the Smarandache sequences, to the convergence property of some infinite series involving the Smarandache type sequences. List of the Contributors The Editors Zhang Wenpeng Yi Yuan Xu Zhefeng Xue Xifeng Ren Ganglian Ding Liping Li Jie Liu Yanni Ma Jinping Li Zhanhu Wang Ting Yang Hai Ren Dongmei Lu Yaming Liu Duansen Li Junzhuang Li Chao Yang Cundian Zhao Jian Wang Nianliang Gao Nan Lou Yuanbing Yang Qianli Yang Mingshun Ren Zhibin Zhao Xiaopeng Wang yongxing Feng Zhiyu Chen Guohui Du Jianli Fu Ruiqin Zhang Shengsheng Huang Wei Ji Yongqiang Zhao Jiantang Li Yansheng Ma Junqing