# 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  proved that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)

H. -E. Richert  proved

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)

Suppose [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Recently Lulu Zhang  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, .

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

 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.

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

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

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

 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).
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