Reciprocals of the Gcd-Sum Functions

Transcription

1 Journal of Integer Sequences, Vol. 4 20, Article.8.3 Reciprocals of the Gcd-Sum Functions Shiqin Chen Experimental Center Linyi University Linyi, , Shandong China shiqinchen200@63.com Wenguang Zhai Department of Mathematics China University of Mining and Technology Beijing, China zhaiwg@hotmail.com Abstract In this paper we study the reciprocals of the gcd-sum function and some related functions and improve some results of Tóth. The harmonic mean of the gcd function is also studied. Introduction Recently, L. Tóth published a paper [3] about the gcd-sum function Pn, called also Pillai s function [2] defined by n Pn gcdk,n. k In this well-written paper, Tóth not only listed many classical results about this function, its analogues and generalizations, but also proved several new results. The aim of this short note is to improve some results in Tóth s paper. This work is supported by the Natural Science Foundation of Beijing Grant No. 200.

2 Let Hn denote the harmonic mean of gcd, n, gcd2, n,..., gcdn, n, namely, n n 2 Hn : n gcdj, n dϕd, j d n where ϕn denotes Euler s function. Theorem 4 of Tóth [3] states that Hn n C log x + C 2 + Ox +ε, 2 where C and C 2 are computable constants, ε > 0 is a small positive constant. As a corollary, he deduced that Hn C x + Ox ε. 3 In this note we shall prove first the following Theorem. We have the asymptotic formula Hn C x + C 3 log x + Olog x 2/3, 4 where C and C 3 are constants. As a corollary of Theorem, we have Corollary 2. Hn n C log x + C 2 + Ox log x 2/3. 5 Remark 3. Tóth [3] studied the average order of Hn/n first and then deduced an asymptotic formula of Hn through it. However in this note we study Hn first, then deduce an asymptotic formula for Hn/n. Now we study the reciprocals of Pn and some other related functions. We first recall the definitions of some functions. An integer d is called a unitary divisor of n if d n and d,n/d, notation d n. The unitary analogue of the function P is defined by P n : n j,n, where j,n max{d N : d j,d n}, which was first introduced in Tóth [4]. This function P A45388 is multiplicative and for every prime power p α α we have P p α 2p α. Let n > be an integer of canonical form n r j pa j j. An integer d is called an exponential divisor of n if d r j pb j j such that b j a j j r, notation d e n. By convention e. The kernel of n is denoted by κn r j p j. 2 j

3 Two positive integers m >, n > have common exponential divisors iff they have the canonical forms n r j pa j j and m r j pb j j with a j,b j j r. The greatest common exponential divisor of n and m is n,m e r j pa j,b j j. By convention define, e and note that,m e does not exist for m >. Now define the function P e n by the relation P e n j,n e. j n κjκn The function P e see Tóth [5] is multiplicative and for any prime power p α α we have P e p α j αp j,α d α p d ϕα/d. 6 Now we recall another analogue of P. Let n > be an integer and we say an integer j is regular mod n if there exists an integer x such that j 2 x j mod n. Tóth [6] introduced the function A76345 Pn : j Reg n gcdj,n, where Reg n { j n : j is regularmod n}. Tóth[6] showed that P is multiplicative and Pn n p n2 /p. 7 The above mentioned functions have been extensively studied and many papers about them have been published. See Tóth [3] and references therein. Theorem 6 of Tóth [3] states that Pn P n Pn P e n where K,K, K,K e are computable constants. In this note we shall prove the following result. Klog x /2 + Olog x /2, 8 K log x /2 + Olog x /2, 9 Klog x /2 + Olog x /2, 0 K e log x + O, 3

4 Theorem 4. Suppose N is a fixed integer, then Pn P n Pn P e n N K j log x /2 j + Olog x /2 N, 2 j0 N Kj log x /2 j + Olog x /2 N, 3 j0 N j0 K j log x /2 j + Olog x /2 N, 4 K e 0 log x + K e + Ox log 2 x, 5 where K j,kj, K j,k e j j 0 are computable constants. Notation. Throughout this paper, ε denotes a sufficiently small positive constant. ζs denotes the Riemann zeta-function. For any real number t, [t] denotes the greatest integer not exceeding t, {t} t [t], and ψt {t} /2. For any complex number z, σ z n d n dz and d z n denotes the generalized divisor function, µn denotes the Möbius function. 2 Proof of Theorem Define for Rs > that D H s : Hnn s. 6 n Since Hn is multiplicative, by Euler s product we get D H s + Hp α p αs. 7 p We evaluate Hp α first for any prime p and α. By we have Hp α p 2α + α α p 2j p 8 j p 2α + p p α p 2j j p 2α + p2α+2 p 2 pp + p p 2α 2α+ + p + p2α p + p 2α+ + + p + p 2α. 4

5 Hence we have which implies that + + Hp α p αs 9 α + p + p 2α p αs α + + p + + p p αs p 2α m α m0 m p m m0 α p αs 2αm + + p m p m p s 2m p s 2m m0 + + p p s p + + s p m p m p s 2m p s 2m m + p s + + p m p m p s 2m p s p s 2m, p s p s + m Hp α p αs 20 α p 2s 2 + p s p s + p + Op 2σ p σ + p σ p σ 3 + Op 2σ 2 + p σ 3 + p 2σ 3 + p 3σ 4, where σ Rs. From 7, 20 and noting m p m p s 2m p s 2m m ζs p p s Rs > 2 we get D H s ζsζs + Gs Rs >, 22 where Gs p s p s + Hp α p αs p α 23 such that if we expand Gs into a Dirichlet series Gs gnn s, 24 n 5

6 then this Dirichlet series is absolutely convergent for Rs > /2. From 22 and 24 we have For σ n we have σ n Hn d n n n n [ x n n] n x n 2 2 σ dgn/d. 25 /n 26 x n 2 ψx n n ψx n n π2 x 6 log x + Olog x 2/3, 2 where in the last step we used the well-known boundsee Walfisz [7] From we have Hn π2 x 6 +O gm n ψx n log x2/3. 27 /m σ n 28 gm m log x 2 log 2/3 x gm + gm log m 2 gm Recall that the Dirichlet series n gnn s is absolutely convergent for Rs > /2. Hence for any U > we have gn U /2+ε. U<n 2U It follows that the infinite series m gmm, m gm, m gm log m are all convergent and that gm m gm m. gm m + Ox 3/2+ε, 29 gm + Ox /2+ε m gm log m, Now Theorem follows from 28 and gm.

7 3 Proof of Theorem 4 In this section we shall prove Theorem 4. We only prove 2 and 5 since the proofs of 2, 3 and 4 are the same. 3. The generalized divisor problem Suppose k 2 k N is a fixed integer. The divisor function d k n denotes the number of ways n can be written as a product of k natural-number factors. It is an important problem in the analytic number theory to study the mean value of the divisor function d k n. It is well-known that d k n are the coefficients of the Dirichlet series ζ k s d k nn s Rs >. n Now suppose z is a fixed complex number and let d z n denote the coefficients of the Dirichlet series ζ z s : d z nn s Rs >, 30 n where ζ z s e z log ζs such that for log s we take the main value log 0. The function d z n is called the generalized divisor function. Suppose A > 0 is an arbitrary but fixed real number and N is an arbitrary but fixed integer. Then uniformly for z A we have d z n x N c j zlog x z j + Oxlog x Rz N, 3 j where the functions c z,,c N z are regular in the region z A. The above result is Theorem 4.9 of Ivić []. 3.2 Proof of 2 Define P n n/pn. Then by Euler s product we have for Rs > that + P p α p αs. 32 n P nn s p α We evaluate P p α α first. The formula 4 of [3] reads Pp α α + p α αp α, which implies that P p α p α α + p α αp α α + + Op. 33 7

8 Inserting 33 into 32 and recalling 2 we get that P nn s ζ /2 sg s, Rs >, 34 n where G s p s /2 + p such that if we expand G s into a Dirichlet series P p α p αs α 35 G s g nn s, 36 n then this Dirichlet series is absolutely convergent for Rs > /2. For the function g we have the trivial estimate g m y /2+ε. 37 m y From 37 we get by partial summation that g m m, g m m y /2+ε 38 and for any fixed constant C that m y m>y m y g m log C m m m g m log C m m + Oy /2+ε. 39 Let e C denote the value of the infinite series in 39. Suppose β is a real number which is not a non-negative integer. By Taylor s expansion we have u β N l0 d β l u l + O u N+, u /2, 40 where d β l l ββ β l + /l!. By the hyperbolic approach, 3 with A z /2, 38 and 39 with y x and 40 8

9 we get for any fixed N that P n d /2 ng m 4 where K j n m x jj +l j,l 0 m x m x g m /m g m /m d /2 n + n x d /2 n + Ox 3/4+ε d /2 n g m x</n x N g m c j /2log x m m /2 j + O x m log x /2 N j +Ox 3/4+ε N x c j /2log x /2 j j m x +Oxlog x /2 N N N x c j /2log x /2 j j +O x l0 N log x /2 j j m x +Oxlog x /2 N N N x c j /2log x /2 j j l0 g m m d /2 j l g m m log m /2 j log x g m log l m m log l x log N+ m log N+ x m x d /2 j l e l log l x +Oxlog x /2 N N x K j log x /2 j + Oxlog x /2 N, j c j /2d /2 j l e l j l0 c j l /2d /2 j+l l e l j N. From 4 we get 2 immediately by partial summation and some easy calculations. 3.3 Proof of 5 9

10 Now we prove 5. Define P e n n/p e n. By Euler s product we have for Rs > that P e nn s + P e p α p αs. 42 p n α Suppose p is a prime. From 6 it is easy to see that P e p p,p e p 2 p 2 + p,p e p 3 p 3 + 2p,P e p α p α + Op α/2 α 4. Hence P e p,p e p 2 P e p α e + p,p p 3 + 2p 2, 43 p α p α + Op α/2 + Op α/2 + Op α/2 α 4. From 43 we get α p αs + α p αs + α P e p α p αs 44 α α2 P e p α p αs P e p α p αs p + s + p p 2s + + 2p 2 p 3s + P e p α p αs α4 p + s + p p 2s + O p 3σ 2 + p α/2 ασ Hence we get α4 p + s + p p 2s + O p 3σ 2 + p 4σ 2 p s p 2s + O p 2σ 2 + p 3σ 2 + p 4σ 2 p s p 2s + O p 2σ 2 + p 4σ 2. p s + α P e p α p αs 45 p 2s p s + O p 2σ 2 + p 2σ + p σ p 2s + p 3s + O p 2σ 2 + p 3σ 0

11 and and p s + p 2s + α P e p α p αs 46 p 2 4s + p 3s + p 2 5s + O p 2σ 2 + p 3σ + p 2σ + p 3s + O p 2σ 2 + p 3σ + p 2σ p s + p 2s p 3s + α p 2 6s + O p 2σ 2 + p 3σ + p 2σ + p 3σ P e p α p αs 47 + O p 2 6σ + p 2σ 2 + p 3σ + p 2σ + p 3σ. We write for σ > + α P e p α p αs 48 p s + p 2s p 3s p s + p 2s p 3s + p 2s p s p 2 4s p 3s p s + p 2s p 3s + From 48 we may write for σ > that n P e nn s ζsζ3s + ζ2s + α α P e p α p αs P e p α p αs G e P s, 49. where G e P s ζ4s + 2 p p s + p 2s p 3s + α P e p α p αs. 50 From 47 we see that if we write the function G e P s into a Dirichlet series G e P s n g e P nn s, 5

12 then this Derichlet series is absolutely for σ > /6. This fact implies that g e P n, g e P n g e P n + Ox 7/6+ε. 52 n n n From 49, 5 and 52 we get P e n xg P e xg P e x x x x x x xζ4 x xζ4 x x ζ4 ζ3 x ζ4 ζ3 n n 2 2 n3 3 x µn 2 n n 2 2 n3 3 x 2 n 3 µn 2 n n 2 2 n3 3 x 2 n 3 g e P g e P g e P x µn 2 n 3 2 n 2 2 n3 3 x 3 n 2 x n 2 x g e P µn 2 g e n 2 n P 53 3 µn 2 n 3 2 µn 2 n 3 2 n 2 x n x n 2 2 n3 3 x n 2 2n O + O x x n 3 n 2 2 g e P log 2 x + Olog 2 x /3 3 ζ4 + O n 2 2 x µn 2 n 3 2 g e P ζ3 + O x g e P + Olog 2 x g e P + Olog 2 x, + Olog 2 x + Olog 2 x + Olog 2 x where we used the following easy estimatesy 2 n 4 ζ4 + Oy 3, 54 n y µnn 3 ζ3 + Oy 2, 55 n y n log y. 56 n y 2

13 Now 5 follows from 53 by partial summation. 4 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. References [] A. Ivić, The Riemann Zeta-Function, John Wiley & Sons, 985. [2] S. S. Pillai, On an arithmetic function, J. Annamalai Univ , [3] L. Tóth, A survey of gcd-sum functions, J. Integer Sequences, 3 200, Article [4] L. Tóth, The unitary analogue of Pillai s arithmetical function, Collect. Math , [5] L. Tóth, On certain arithmetic functions involving exponential divisors, Annales Univ. Sci. Budapest., Sect. Comp , [6] L. Tóth, A gcd-sum function over regular integers mod n, J. Integer Sequences , Article [7] A. Walfisz, Weylsche Exponentialsummen in der neueren Zahlentheorie, Berlin, Mathematics Subject Classification: Primary A25; Secondary N37. Keywords: gcd-sum function, regular integers, harmonic mean, unitary divisor. Concerned with sequences A08804, A45388 and A Received February 3 20; revised versions received May 5 20; June 28 20; August Published in Journal of Integer Sequences, September Return to Journal of Integer Sequences home page. 3