Printer Friendly

On the mean value of log [??](n).

[section] 1. Introduction and main results

Let n > 1 be an integer. Consider the integers m for which there exists an integer x such that [m.sup.2]x = m(mod n). Let [??](n) = {m: 1 [less than or equal to] m [less than or equal to] n, m is regular(mod n)}. This function is multiplicative and [??]([p.sup.[gamma]]) = [psi]([p.sup.[gamma]]) + 1 = [p.sup.[gamma]] - [p.sup.[gamma]-1] + 1 for every prime power [p.sup.[gamma]] ([gamma] [greater than or equal to] 1), where [psi](n) is the Euler function (see [1]).

The mean value of the function [??](n) was considered in [2], [4]. One has,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where A = [[PI].sub.p](1 - 1 / [p.sup.2](p+1)) = [zeta](2) [[PI].sub.p] (1 - 1/[p.sup.2] - 1/[p.sup.3] + 1/[p.sup.4]) [approximately equal to] 0.8815 is the so called quadratic class-number constant.

More exactly, V. S. Joshi [2] proved

[summation over (n [less than or equal to] x)] [??](n) = 1/2 A[x.sup.2] + R(x), (1)

where R(x) = O(x [log.sup.3] x). This was improved into R(x) = O(x [log.sup.2] x) in [3], and into R(x) = O(x log x) in [6]. The [OMEGA]-estimate R(x) = [[OMEGA].sub.[+ or -]](x [square root of log log x]) was also proved in [6].

Laszlo Toth [1] proved the following three results:

[summation over (n [less than or equal to] x)] [??](n)/[psi](n) = 3/[[pi].sup.2]x + 0([log.sup.2] x), (2)

[summation over (n [less than or equal to] x)] [psi](n)/[??](n) = Bx + 0([(log x).sup.5/3] [(log log x).sup.4/3]), (3) [??](n)

[summation over (n [less than or equal to] x)] 1/[??](n) = [C.sub.1] log x + [C.sub.2] + O([log.sup.9] x / x), (4)

where [C.sub.1] and [C.sub.2] are constants,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

In this paper, we shall prove a result about the mean value of log [??](n). Our main result is the following

Theorem. We have

[summation over (n [less than or equal to] x)] log [??](n) = x log x + Ex + O([x.sup.1/2] [log.sup.3/2] x), (5)

where

E = [summation over p] (1 - [p.sup.-1]) [[infinity].summation over [alpha] = 2] [p.sup.-[alpha]] log(1 - [p.sup.-1] + [p.sup.-[alpha]]).

Notations. Throughout this paper, [epsilon] > 0 denotes a small positive constant.

[section] 2. Proof of the theorem

In order to prove our theorem, we need the following lemmas, which can be found in Ivic [5]. From now on, [zeta](s) denotes the Riemann-zeta function.

Lemma 1. Let T [greater than or equal to] 2 be a real number, then we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where [gamma] is the Euler constant.

Lemma 2. For t [greater than or equal to] [t.sub.0] [greater than or equal to] 2, we have uniformly for [sigma] that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Proof of Theorem.

Let f (n): = [[??].sup.u](n), where u is a fixed complex number with [absolute value of u] [less than or equal to] 1/4. Then for every prime power [p.sup.[alpha]],

f([p.sup.[alpha]]) = [[??].sup.u]([p.sup.[alpha]]) = [([p.sup.[alpha]] - [p.sup.[alpha]-1] + 1).sup.u] = [p.sup.[alpha]u] [(1 - [p.sup.-1] + [p.sup.-[alpha]]).sup.u]. (6)

Since f(n) is multiplicative, by the Euler product we get for Rs > 1 that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)

Write G(s, u) = [[summation].sup.[infinity].sub.n=1] g(n)/[n.sup.s] (Rs > 1). It is easy to see that this infinite series is absolutely convergent in the range Rs > Ru, which implies that

G(s,u) = O[epsilon](1), Rs [greater than or equal to] Ru + [epsilon]. (9)

Taking T = [x.sup.2]. By Perron's formula we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Then by the residue theorem we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (10)

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (11)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (12)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (13)

Denote s = [sigma] + it, u = Ru + iv, where [absolute value of u] [greater than or equal to] 1/4. Form (9) we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (14)

With the partial summation and (14), we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (15)

where B(t) = [[integral].sup.t.sub.0] [absolute value of [zeta](1/2 + iw)] dw.

With Lemma 1 and Cauchy's inequality, we get (for t [greater than or equal to] 2)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (16)

With (15) and (16), we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (17)

With Lemma 2, (9) and (13), we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Since the integrands in the above integrals above are monotone, we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (18)

Similarly, we have

[[integral].sub.1] << 1.. (19)

By (10)-(13) and (17)-(19), we obtain

[summation over n [less than or equal to] x ] [[??].sup.u](n) = G(1 + u,u) / 1 + u x [x.sup.1+u] + O ([x.sup.1/2+Ru] [log.sup.3/2] x) (20

By differentiating (20) term by term, we derive

[summation over n [less than or equal to] x ] [[??].sup.u](n) log [??](n) = H'(u)[x.sup.1+u] + H(u)[x.sup.1+u] log x + O([x.sup.1/2+Ru] [log.sup.3/2] x), (21)

where H(u): = G(1+u'u)/1+u.

Letting u = 0 in (21), we get

[summation over n [less than or equal to] x ] log [??](n) = H(0) x log x + H'((0)x + O([x.sup.1/2][log.sup.3/2] x). (22)

Now we evaluate H(0) and H'(0). According to (8), we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (23)

which implies immediately that H(0) = 1.

Taking the logarithm derivative from both sides of (23) we get that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (24)

which together with H(1) = 1 gives

H'(0) = [summation over p] (1 - [p.sup.-1]) [[infinity].summation over [alpha]=2] [p.sup.-[alpha]] (log - [p.sup.-1] + [p.sup.-[alpha]]). (25)

Now Theorem follows from (22), (24) and (25).

References

[1] Laszlo Toth (Pecs, Hungary), Regular integers modulo n, Annales Univ. Sci. Budapest., Sect. Comp., 29(2008), 263-275.

[2] V. S. Joshi, Order-free integers (mod m), Number theory (Mysore, 1981), Springer, Lecture Notes in Math., 938(1982), 93-100.

[3] Zhao Hua Yang, A note for order-free integers (mod m), J. China Univ. Sci. Tech. 16(1986), No. 1, 116-118.

[4] S. Finch, Idempotents and nilpotents modulo n, 2006, Preprint available online at http://arxiv.org/abs/math. NT/0605019v1.

[5] A. Ivic, The Riemann Zeta-function, University of Belgrade Yugoslavia, 1985.

[6] J. Herzog, P. R. Smith, Lower bounds for a certain class of error functions, Acta Arith., 60(1992), 289-305.

Lixia Li and Heng Liu

School of Mathematical Sciences, Shandong Normal University, Jinan 250014, P.R.China

E-mail: li09lixia25@163.com sglheng@163.com
COPYRIGHT 2009 American Research Press
No portion of this article can be reproduced without the express written permission from the copyright holder.
Copyright 2009 Gale, Cengage Learning. All rights reserved.

Article Details
Printer friendly Cite/link Email Feedback
Author:Li, Lixia; Liu, Heng
Publication:Scientia Magna
Article Type:Report
Geographic Code:9CHIN
Date:Dec 1, 2009
Words:1213
Previous Article:Euler-Savary formula for the planar homothetic motions.
Next Article:Semigroup of continuous functions and Smarandache semigroups.
Topics:

Terms of use | Copyright © 2017 Farlex, Inc. | Feedback | For webmasters