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A short interval result for the extension of the exponential divisor function.

[section]1. Introduction

In 1972, M. V. Subbarao [3] established the definition of exponential divisor: Let n > 1 be an integer of canonical from [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. The integer [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is called an exponential divisor of n if [b.sub.i]|[a.sub.i] for every i [member of] {1, 2, ..., s}, notation: d[|.sub.e]n. By convention 1[|.sub.e]1. Besides, he also studied the mean value problem of exponential divisor function [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and obtained:

[summation over (n [less than or equal to] x)][[tau].sup.(e)](n) = Ax + E(x),

where E(x) = O([x.sup.1/2]).

J. Wu [5] improved the result of M. V. Subbarao and obtained

[summation over (n [less than or equal to] x)][[tau].sup.(e)](n) = Ax + B[x.sup.1/2] + O([x.sup.2/9]log x),

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

M. V. Subbarao [3] also proved that for any integers r, we have the estimate

[summation over (n [less than or equal to] x)][([[tau].sup.(e)](n)).sup.r] ~ [A.sub.r]x,

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

L. TOth [1,2] proved

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

here [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is a polynomial of degree [2.sup.r] - 2 of t, [u.sub.r] = [2.sup.r + 1]-1/[2.sup.r + 2] + 1.

Similar to the generalization from d(n) to [d.sup.k](n), we extended [[tau].sup.(e)](n) and established a definition as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Obviously [[tau].sup.2](e)(n) = [[tau].sup.(e)](n). In this paper we studied the case of k = 3 which means to study the properties of [([[tau].sup.(e).sub.3](n)).sup.2] and obviously [([[tau].sup.(e).sub.3](n)).sup.2] is a multiplicative function. The aim of this paper is to study the short interval case of [([[tau].sup.(e).sub.3](n)).sup.2] and prove the following theorem.

Theorem 1.1. If [x.sup.1/5 + 2[epsilon]] < y [less than or equal to] x, then

[summation over (x < n [less than or equal to] x + y)][([[tau].sup.(e).sub.3](n)).sup.2] = [c.sub.1]y + O(yx - ii + 3/2[epsilon] + x). (1)

where [c.sub.1] = [Res.sub.s = 1]F(s) and F(s);= [[summation].sup.[infinity].sub.n = 1] [([[tau].sup.(e).sub.3](n)).sup.2]/[n.sup.s].

Notations. Throughout this paper, [epsilon] always denotes a fixed but sufficiently small positive constant. We assume that 1 [less than or equal to] a [less than or equal to] b are fixed integers, and we denote by d(a, b; k) the number of representations of k as k = [n.sub.1]a[n.sup.2]b, where [n.sub.1], [n.sup.2] are natural numbers, that is,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] will be used freely.

[section]2. Proof of the theorem

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

Lemma 2.1. Suppose s = [sigma] + it is a complex number (Rs > 1), then

F(s): = [[summation].sup.[infinity].sub.n = 1][([[tau].sup.(e).sub.3](n)).sup.2]/[n.sup.s] = [zeta](s)[[zeta].sup.2](2s)/[[zeta].sup.3](5s)G(s),

where the Dirichlet series G(s); = [[summation].sup.[infinity].sub.n = 1] g(n)/[n.sup.s][infinity] is absolutely convergent for Rs = [sigma] > 1/6.

Proof. Here [[tau].sup.(e).sub.3](n) is multiplicative and by Euler product formula we have for [sigma] > 1 that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (2)

Now we write G(s): = [[summation].sup.[infinity].sub.n = 1]g(n)/[n.sup.s][infinity]. It is easily seen the Dirichlet series is absolutely convergent for Rs = [sigma] > 1/6.

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 and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Then we have

B(x, y; k, [epsilon]) [much less than] y[x.sup.-[epsilon]] + [x.sup.1/2k + 1]logx. (3)

Proof. This lemma is very important when studying the short interval distribution of 1-free number; see for example [4].

Let [a.sub.1](n), [a.sup.2](n), [a.sup.3](n) and [a.sub.4](n) be arithmetic functions defined by the following Dirichlet series (for Rs > 1):

[[infinity].summation over (n = 1)][a.sub.1](n)/[n.sup.s] = [zeta](s)G(s). (4)

[[infinity].summation over (n = 1)][a.sub.2](n)/[n.sup.2s] = [[zeta].sup.8](2s). (5)

[[infinity].summation over (n = 1)][a.sub.3](n)/[n.sup.4s] = [[zeta].sup.-9](4s). (6)

[[infinity].summation over (n = 1)][a.sub.4](n)/[n.sup.5s] = [[zeta].sup.-27](5s). (7)

Lemma 2.3. Let [a.sub.1](n) be an arithmetic function defined by (4), then we have

[[summation].sub.n [less than or equal to] x][a.sub.1](n) = Cx + O([x.sup.1/6 + [epsilon]]), (8)

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

Proof. Using lemma 1.1, it is easy to see that

[summation over (n [less than or equal to] x)][absolute value of g(n)] [much less than] [x.sup.1/6 + [epsilon]]

Therefore from the definition of g(n) and (4), it follows that

[summation over (n [less than or equal to] x)][a.sub.1](n) = [[summation].sub.mn [less than or equal to] x]g(n) (9)

= [summation over (n [less than or equal to] x)]g(n)[summation over (m [less than or equal to] x/n1)] (10)

= [summation over (n [less than or equal to] x)]g(n)(x/n + O(1)) (11)

= Cx + O([x.sub.1/6 + [epsilon]]), (12)

and C = [Res.sub.s = 1][zeta](s)G(s).

Next we prove our Theorem. From lemma 2.3 and the definition of [a.sub.1](n), [a.sup.2](n), [a.sup.3](n) and [a.sub.4](n), we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (13)

So we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (14)

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (15)

In view of lemma 3,

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

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

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (17)

Similarly we have

[summation over 3] [much less than] [yx.sup.- [member of] /2] + [x.sup.1/5 + 3/2[epsilon]], [summation over 4] [much less than] [yx.sup.- [member of] /2] + [x.sup.1/5 + 3/2[epsilon]]. (18)

Now our theorem follows from (9)-(14).

Acknowledgements. The author deeply thank the referee for this careful reading of the manuscript and many valuable suggestions.

This work is supported by Natural Science Foundation of Jiangxi province of China (Grant Nos. 2012ZBAB211001) and Natural Science Foundation of Jiangxi province of China (Grant Nos. 20132BAB2010031).

References [1] Laszlo Toth, An order result for the exponential divisor function, Publ. Math. Debrean. 71 (2007), No. 1-2, 165-171.

[2] Laszlo Toth, On certain arithmetic function involing exponential divisors, II. Annales Univ. Sci. Budapest. Sect. Comp., 27 (2007), 155-156.

[3] M. V. Subbarao, The theory of arithmetic functions (Proc. Conf., Western Michigan Univ., Kalamazoo, Mich., 1971), Lecture Notes in Math., Springer, Berlin, (1972), 251.

[4] W. G. Zhai, Square-free numbers as sum of two squares, Number Theory: Tradition and modernization, Spring, New York, (2006), 219-227.

[5] J. WU, Problem of exponential divisors and exponentially square-free integers, J. Theor. Nombres Bordeaux, 7(1995), No. 1, 133-141.

Li Yang

Department of Mathematics, Nanchang University

Nanchang, Jiangxi 330031, P. R. China

E-mail: yangli46@ncu.edu.cn
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Date:Sep 1, 2013
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