Some Arithmetic Functions Involving Exponential Divisors

Transcription

1 Journal of Integer Sequences, Vol , Article Some Arithmetic Functions Involving Exponential Divisors Xiaodong Cao Department of Mathematics and Physics Beijing Institute of Petro-Chemical Technology Beijing, 0267 P. R. China caoxiaodong@bipt.edu.cn Wenguang Zhai School of Mathematical Sciences Shandong Normal University Jinan, Shandong P. R. China zhaiwg@hotmail.com Abstract In this paper we study several arithmetic functions connected with the exponential divisors of integers. We establish some asymptotic formulas under the Riemann hypothesis, which improve previous results. We also prove some asymptotic lower bounds. Introduction and results Suppose n > is an integer with prime factorization n p a p ar r. An integer d is called an exponential divisor e-divisor of n if d p b p br r with b j a j j r, which is denoted by the notation d e n. For convenience e. The properties of the exponential divisors attract the interests of many authorssee, for example, [4, 6, 7, 9,, 2, 4, 5, 6, 9, 20, 2, 22].

2 An integer n p a p ar r is called exponentially squarefree e-squarefree if all the exponents a,,a r are squarefree. The integer is also considered to be an e-squarefree number. Suppose f and g are two arithmetic functions and n p a p ar r. Subbarao [7] first introduced the exponential convolution e-convolution by f gn b c a b rc ra r fp b p br r gp c p cr r, which is an analogue of the classical Dirichlet convolution. The e-convolution is commutative, associative and has the identity element µ 2, where µ is the Möbius function. Furthermore, a function f has an inverse with respect to iff f 0 and fp p r 0 for any distinct primes p,,p r. The inverse of the constant function fn with respect to is called the exponential analogue of the Möbius function and it is denoted by µ e. Hence d en µe d µ 2 n for n, µ e, and µ e n µa µa r for n p a p ar r >. Note that µ e n or 0, according as n is e-squarefree or not. An integer d is called an exponential squarefree exponential divisor e-squarefree e- divisor of n if d p b p br r with b j a j j r and b,,b r are squarefree. Observe that the integer is an e-squarefree and it is not an e-divisor of n >. Let t e denote the number of e-sqaurefree e-divisors of n. Now we introduce some notation for later use. As usual, let µn and ωn denote the Möbius function, and the number of distinct prime factors of n respectively. If t is real, then {t} denotes the fractional part of t,ψt {t} /2. Throughout this paper, ε is a small fixed positive constant and m M means that cm m CM for some constants 0 < c < C. For any fixed integers a and b, define the function da,b;n : m a mb 2 n and let a,b;t denote the error term in the asymptotic formula of the sum da,b;n. Many authors have studied the properties of the above three functions; see, for example, [4, 7, 5, 7, 9, 20, 22]. Tóth [20] proved that µ e n A x + x /2 exp c log x, where 0 < < 9/25 and c > 0 are constants and A : mµ e µk µk +. p k p k2 Tóth [20] proved that if the Riemann hypothesis RH is true, then µ e n A x + Ox 9/202+ε. 2 Subbarao [7] and Wu [22] studied the asymptotic properties of the sum µe n. Tóth [20] proved that if RH is true, then µ e n B x + Ox /5+ε, 3 2

3 where where B : p + α4 µ 2 a µ 2 a p a. For the function t e n, Tóth [20] proved the asymptotic formula t e n C x + C 2 x /2 + Ox /4+ε, 4 C : p C 2 : ζ 2 p + 2 ωk 2 ωk, k2 + p k 2 ωk 2 ωk 2 ωk ωk 3. k4 Pétermann [3] proved the formula 4 with a better error term Ox /4, which is the best unconditional result up to date. In order to further reduce the exponent /4, we have to know more information about the distribution of the non-trivial zeros of the Riemann zeta-function. In this short paper we shall prove the following theorem. Theorem. If RH is true, then µ e n A x + O p k x ε, 5 µ e n B x + B 2 x 5 + Ox ε, 6 t e n where B 2 is a computable constant. Remark. Numerically, we have C x + C 2 x /2 + O x ε, , , < /5, < /4. Let µ ex, µ e x, t ex denote the error terms in 5, 6 and 7 respectively. We have the following result. Theorem 2. We have µ ex Ωx /4, 8 µ e x Ωx /8, 9 t ex Ωx /6. 0 3

4 2 Some generating functions In this section we shall study the generating Dirichlet series of the functions µ e n, µ e n and t e n, respectively. We consider only µ e n and t e n, since Tóth [20] already proved the following formula, which is enough for our purpose, n µ e n ζs Us Rs >, ζ 2 2s where Us : un n is absolutely convergent for Rs > /5. We first consider the function µ e n. The function µ e is multiplicative and µ e p a µa for every prime power p a, namely for every prime p, µ e p,µ e p 2,µ e p 3,µ e p 4 0,µ e p 5,µ e p 6,µ e p 7,µ e p 8 0,µ e p 9 0,µ e p 0,µ e p,. Hence by the Euler product we have for Rs > that n µ e n p + m µm p ms Applying the product representation of Riemann zeta-function. 2 ζs p + p s + p 2s + p p s Rs >, 3 we have for Rs > ζsζ5s p p s p 5s. 4 Let f µ z : + µm z m 5 e m + z + z 2 + z 3 + z 5 + z 6 + z 7 + z 0 + z + For z <, it is easy to verify that µm z m. m2 where f µ z z z 5 6 e + z + z 2 + z 3 + z 5 + z 6 + z 7 + z 0 + z + µm z m z z 5 + z 6 z 4 z 8 + z 9 + m2 c m z m, m2 c m : µm µm µm 5 + µm 6. m 2 4

5 Furthermore, we have f µ z z z 5 z 4 z 8 7 e z 4 z 8 + z 9 + c m z m + z 4 + z z 8 + z z 9 + m2 C m z m. m2 We get from 2, 3, 5 and 7 by taking z p s that n µ e n ζsζ5s V s Rs >, 8 ζ4sζ8s where V s : vn n is absolutely convergent for Rs >. 9 We now consider the function t e n, which is also multiplicative and t e p a 2 ωa for every prime power p a. Therefore for every prime p, t e p,t e p 2 t e p 3 t e p 4 t e p 5 2,t e p 6 4,t e p 7 t e p 8 t e p 9 2,t e p 0 4,. By the Euler product we have for Rs > that Let f t ez : + n t e n p + m 2 ωm. 9 p ms 2 ωm z m 20 m + z + 2z 2 + 2z 3 + 2z 4 + 2z 5 + 4z 6 + 2z 7 + 2z 8 + 2z 9 + 4z 0 + 2z + For z <, it is easy to check that and f t ez z z 2 z 6 2 z 4 2z 7 2z 8 + 2z 9 + m0 c 2 m z m m2 2 ωm z m. f t ez z z 2 z 6 2 z 4 2 z 4 2z 7 2z 8 + 2z 9 + c 2 m z m + z 4 + z 8 + 2z 7 + m8 C 2 m z m. 5 m0

6 Combining 3 and 9 2 we get n t e n ζsζ2sζ2 6s Ws Rs >, 22 ζ4s where Ws : wn n is absolutely convergent for Rs >. 7 3 Proof of Theorem 2 We see from that the generating Dirichlet series of the function µ e n is Us, ζ 2 2s which has infinitely many poles on the line Rs, whence the estimate 8 follows. From 4 the expression 8 we see that the generating Dirichlet series of the function µ e n is ζsζ5s V s, which has infinitely many poles on the line Rs, whence the estimate 9 ζ4s 8 follows. Via 22, we get the estimate 0 with the help of Theorem 2 of Küleitner and Nowak [8] or by Balasubramanian, Ramachandra and Subbarao s method in []. 4 The Proofs of 5 and 7 We follow the approach of Theorem in Montgomery and Vaughan [0]. Throughout this section, we assume RH. We first prove 5. It is well-known that the characteristic function of the set of squarefree integers is µ 2 n µn d 2 n µd, 23 ζs We write x : µ 2 d x ζ2 : Dx x ζ2. 24 Define the function a n by a n µ 2 mµd, 25 it is easy to see that md 2 n ζs ζ 2 2s n µ 2 n ζ2s n a n, Rs >, 26 Thus T x : a n md 2 x µ 2 mµd d x 2 µdd x d

7 Suppose < y < x /2 is a parameter to be determined. We now write T x in the form T x S x + S 2 x, 28 where S x x µdd, 29 d 2 d y and S 2 x y<d x 2 x µdd d 2 md 2 x d>y µ 2 mµd. 30 We first evaluate S x. From 24 we get S x x x µd d 2 ζ2 + d 2 d y x µd + x µd. ζ2 d 2 d 2 d y d y 3 To treat S 2 x, we let g y s ζ s d y µd, s σ + it, 32 ds so that for Rs σ > g y s d>y µd d s. 33 Hence g y 2sζs ζ2s n b n, 34 for σ >, where b n md 2 n d>y µ 2 mµd 35 Let < σ < 2, δ ε. Assume RH, from 4.25 in Titchmarsh [8] we have 2 0 d y µd d s ζ s + O y 2 σ+δ t δ

8 In addition, RH implies that ζs t δ + and ζ s t δ + uniformly for σ > 2 + δ and s > ε, and n y µn y 2 +δ. From 32, 33 and 36, we have Thus g y 2s y 2 t δ +, σ 2 + δ. g y 2s ζs ζ2s y 2 t 3δ +, σ + δ, s > ε From 30, 34, 35 and Perron s formula[8], Lemma 3.9, we obtain S 2 x b n 2πi +ε+ix 2 +ε ix 2 g y 2s ζs ζ2s xs s ds + Ox δ, 38 since bn n δ by a divisor argument. If we move the line of integration to σ + δ, then 2 by the residue theorem +ε+ix 2 g y 2s ζs 2πi +ε ix ζ2s xs s ds 39 2 I + I 2 I 3 + Res g y2s ζs s ζ2s xs s, where I +ε+ix 2 g y 2s ζs 2πi ζ2s xs s ds,i 2 2πi 2 +δ+ix2 I 3 2πi +ε ix 2 g y 2s ζs ζ2s xs s ds. 2 +δ ix2 From 37 it is not difficult to see that Combining 38 40, we get 2 +δ+ix2 g y 2s ζs ζ2s xs s ds, 2 +δ ix2 I j y 2 x 2 +8δ, j, 2, S 2 x x µd + O ζ2 d 2 d>y y 2 x 2 +ε + x ε. 4 Finally, combining 27, 28, 3 and 4 we obtain T x x ζ x µd + O x d 2 2 +ε y 2 + x ε. 42 d y In [5], Jia proved the estimate u u ε. Inserting this estimate into 42 and on taking y x57, 0 we get T x x ζ O x ε. 43 8

9 The asymptotic formula 5 follows form and 43 by the well-known convolution method. Now we prove the asymptotic formula 7. We define the functions a 2 n by the following identity Hence ζsζ2s ζ4s n a 2 n d 4 mn a 2 n, Rs >. 44 µdd, 2; m. Thus by the same approach as 42, we easily get that for y x 4 T 2 x : a 2 n 45 ζ2 ζ4 x + ζ 2 ζ2 x 2 + d y µd, 2; x + O x d 4 2 +ε y x ε. Graham and Kolesnik [2] proved that, 2;u u ε. Inserting this estimate into 45 and on taking y 5469, 267 we get T 2 x a 2 n ζ2 ζ4 x + ζ 2 ζ2 x 2 + O x ε. 46 The asymptotic formula 7 follows from 22, 44 and 46 by the convolution method. 5 Proof of 6 Throughout this section we assume RH. Define the function a 3 n by ζsζ5s ζ4s n a 3 n, Rs >. 47 Similar to 45 we have for some y x 4 that T 3 x : a 3 n ζ5 ζ4 x + ζ 5 ζ 4 5 x 5 + d y µd, 5; x + O x d 4 2 +ε y x 5 +ε y x ε. Unfortunately, we can not improve Tóth s exponent in 3 by the above formula even if we 5 use the conjectural bound, 5;t t /2+ε. So we use a different approach to prove 6. 9

10 Let q 4 n denote the characteristic function of the set of 4-free numbers, then We write n q 4 n 4 x : n ζs, Rs >. ζ4s q 4 n x ζ4. 48 From 47 and 48, it follows from the Dirichlet hyperbolic approach that for some y x5, B 3 x a 3 n q 4 m 49 md 5 x q 4 m + q 4 m d y d 5 x ζ4 d y + x 4 d 5 d y d 5 + x 5 q 4 mψ Applying partial summation formula, we get q 4 m m 5 q 4m y x 5 y x 5 x q 4 m + q 4 m + y<d x m 5 q 4 m y q m 4 m 5 y 5 x m 5 x x 5 4ζ4 m t q 4m dt t + ζ4 4t x 4 5 y 4 + ψy q 4 m. dt + 5 x 4 t dt. In addition, we have by Euler-Maclaurin formula d ζ5 5 4y 4 ψyy 5 + Oy 6. 5 d y If RH is true, Graham and Pintz [3] showed that 50 4 x Ox ε, 52 0

11 and so x 4 t dt 4 t dt 4 t dt + O x 4 t dt 53 x ε y 3 38 Combining and taking on y x 3/93, we obtain T 3 x q 4 m 54 md 5 x ζ5 ζ4 x + 5 q 4 mψ where we used the trivial estimate and where 4 t dt 4ζ4 + O x m 5 x 5. x ε y xy 6 ζ5 ζ4 x + C 3x /5 + Ox ε y xy 5, ζ5 ζ4 x + C 3x /5 + Ox ε, q 4 mψ C 3 : 5 x m 5 x 4 t dt 4ζ4. Now the asymptotic formula 6 follows from 8, 47, 49 and 54 by the convolution approach. 6 Acknowledgments The authors express their gratitude to the referee for a careful reading of the manuscript and many valuable suggestions, which highly improve the quality of this paper. This work is supported by National Natural Science Foundation of China Grant No and Research Award Foundation for Young Scientists of Shandong Province No. BS2009SF08. References [] R. Balasubramanian, K. Ramachandra and M. V. Subbarao, On the error function in the asymptotic formula for the counting function of k-full numbers, Acta Arith , 07 8.

12 [2] S. W. Graham and G. Kolesnik, On the difference between consecutive squarefree integers, Acta Arith , [3] S. W. Graham and J. Pintz, The distribution of r-free numbers, Acta Math. Hungar , [4] P. Jr. Hagis, Some results concerning exponential divisors, Internat. J. Math. Math. Sci. 988, [5] Chaohua Jia, The distribution of square-free numbers, Sci. China Ser. A , [6] I. Kátai and M. V. Subbarao, On the iterates of the sum of exponential divisors, Math. Pannon , [7] I. Kátai and M. V. Subbarao, On the distribution of exponential divisors, Ann. Univ. Sci. Budapest. Sect. Comput , [8] M. Küleitner and W. G. Nowak, An omega theorem for a class of arithmetic functions, Math. Nachr , [9] L. Lucht, On the sum of exponential divisors and its iterates, Arch. Math. Basel, , [0] H. L. Montgomery and R. C. Vaughan, On the distribution of square-free numbers, in Recent Progress in Analytic Number Theory, Vol., Academic Press, 98, pp [] W. G. Nowak and M. Schmeier, Conditional asymptotic fomula for a class of arithmetic functions, Proc. Amer. Math. Soc , [2] Y.-F. S. Pétermann and J. Wu, On the sum of exponential divisors of an integer, Acta Math. Hungar , [3] Y.-F. S. Pétermann, Arithmetical functions involving exponential divisors: Note on two paper by L. Tóth, Ann. Univ. Sci. Budapest. Sect. Comput., to appear. [4] J. Sándor, A note on exponential divisors and related arithmetic functions, Sci. Magna 2005, [5] A. Smati and J. Wu, On the exponential divisor function, Publ. Inst. Math. Beograd N.S., 6 997, [6] E. G. Straus and M. V. Subbarao, On exponential divisors, Duke Math. J , [7] M. V. Subbarao, Oome arithmetic convolutions, in The Theory of Arithmetic Functions, Lecture Notes in Mathematics, Vol. 25, Springer, 972, pp [8] E. C. Titchmarsh, The Theory of the Riemann Zeta-Function revised by D. R. Heath- Brown, Oxford University Press,

13 [9] L. Tóth, On certain arithmetic functions involving exponential divisors, Ann. Univ. Sci. Budapest. Sect. Comput , [20] L. Tóth, On certain arithmetical functions involving exponential divisors II, Ann. Univ. Sci. Budapest. Sect. Comput , [2] L. Tóth, An order result for the exponential divisor function, Publ. Math. Debrecen , [22] Jie Wu, Problème de diviseurs exponentiels et entiers exponentiellement sans facteur carré, J. Théor. Nombres Bordeaux 7 995, Mathematics Subject Classification: Primary N37. Keywords: exponential divisor, error term, asymptotic lower bound. Received January ; revised version received March Published in Journal of Integer Sequences, March Return to Journal of Integer Sequences home page. 3