Printer Friendly

A short interval result for the function [a.sup.2](n).

[section] 1. Introduction

We define a(n) to be the number of nonisomorphic Abelian groups with n elements. The properties of a(n) were investigated by many authors.

P. Erdos and G. Szekeres t1] first proved that

[summation.over n[less than or equal to]x] a(n)= [c.sub.1]x + O ([x.sup.1/2]),(1)

Kendall and Rankin [2] proved that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)

H. -E. Richert [3] proved

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)

Suppose [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Recently Lulu Zhang [4] proved that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)

where [c.sub.j] (j = 4, 5, 6, 7) are computable constants.

In this short paper, we shall prove the following short interval result.

Theorem 1.1. If [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)

Notations 1.1. Throughout this paper, [member of] always denotes a fixed but sufficiently small positive constant. If 1 [less than or equal to] a [less than or equal to] b [less than or equal to] c are fixed integers, we define

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

[section]2. Proof of the theorem

In order to prove our theorem, we need the following lemmas.

Lemma 2.1. Suppose s is a complex number (Rs > 1), then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)

where G(s) can be written as a Dirichlet series [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], which is absolutely convergent for Rs [greater than or equal to] 1/5.

Proof. The function [a.sup.2](n) is multiplicative, So by Euler product formula, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Now we write [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. It is easily seen the Dirichlet series is absolutely convergent for Rs [greater than or equal to] 1/5.

Lemma 2.2. Let k [greater than or equal to] 2 be a fixed integer, 1 < y [less than or equal to] x be large real numbers. Then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Proof. This lemma is often used when studying the short internal distribution of 1-free numbers; see for example, [5].

Lemma 2.3. Let [G.sub.0](s) = [zeta](s)G(s),[f.sub.0](n) be the arithmetic function defined by

[[infinity].summation over n=1)] [f.sub.0](n)/[n.sup.s] = [zeta](s)G(s,)

then we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)

where A = [Res.sub.s=1][zeta](s)G(s).

Proof. From Lemma 2.1 the infinite series [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] converges absolutely for [sigma] > 1/5, it follows that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Therefore from the definition of g(n) and [f.sub.0] (n), it follows that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where A= [Res.sub.s=1][zeta](s)G(s).

Now we prove our theorem. From Lemma 2.3 and convolution method, we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

So we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

In view of Lemma 2.3,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (9)

where C = [Res.sub.s=1]F(s).

By Lemma 2.2 with k = 2 we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (10)

Similarly we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (11)

Now our theorem follows from (8)-(11).

References

[1] P. Erdos and G. Szekeres, Uber die AnZahl der Abelschen Gruppen gegebener Ordnung unduber ein verwandtes zahlentheoretisches Problem, Acta Sci. Math.(Szeged) 7(1935), 95-102.

[2] D. G. Kendall and R. A. Rankin, On the number of abelian groups of a given order, Quart J. Oxford, 18(1947), 197-208.

[3] H. E. Richert, Uber die Anzahl Abelscher Gruppen gegebener Ordnung I, Math. Zeitschr, 56(1952), 21-32.

[4] Lulu Zhang, On the mean value of a2(n), Scientia Magna, 4(2008), No.4, 15-17.

[5] M. Filaseta, O. Trifonov, The distribution of square full numbers in short intervals, Acta Arith., 67(1994), No. 4, 323-333.

Hua Wang ([dagger]) , Jing Li ([double dagger]) and Jian Wang (#)

([dagger] [double dagger]) School of mathematical Sciences, Shandong Normal University, Jinan, 250014, Shandong, P. R. China

(#) SDU Electric Power Technology Co. Ltd, Jinan, 250061, Shandong, P. R. China

E-mail: xnn22wdtx@126.com ljlijing12345@163.com wangjian@sdu.edu.cn

(1) This work is supported by Natural Science Foundation of China (Grant No:11001154), and Natural Science Foundation of Shandong Province(Nos:BS2009SF018, ZR2010AQ009).
COPYRIGHT 2012 American Research Press
No portion of this article can be reproduced without the express written permission from the copyright holder.
Copyright 2012 Gale, Cengage Learning. All rights reserved.

Article Details
Printer friendly Cite/link Email Feedback
Author:Wang, Hua; Li, Jing; Wang, Jian
Publication:Scientia Magna
Date:Sep 1, 2012
Words:756
Previous Article:Estimate of second Hankel determinant for certain classes of analytic functions.
Next Article:A new refinement of the inequality [summation] sin A/2 [less than or equal to] [square root of 4r+r/2r].

Terms of use | Privacy policy | Copyright © 2019 Farlex, Inc. | Feedback | For webmasters