關(guān)于Smarandache-Type可乘函數(shù)的方程

張小蹦,田清

(1.西安郵電學(xué)院應(yīng)用數(shù)理系,陜西西安 710121;2.西北大學(xué)數(shù)學(xué)系,陜西西安 710127)

關(guān)于Smarandache-Type可乘函數(shù)的方程

張小蹦1,2,田清2

(1.西安郵電學(xué)院應(yīng)用數(shù)理系,陜西西安 710121;2.西北大學(xué)數(shù)學(xué)系,陜西西安 710127)

研究了一類包含Smarandache-Type可乘函數(shù)Fk(n)與Gk(n)的無(wú)窮級(jí)數(shù)及其算術(shù)性質(zhì),并利用初等方法及歐拉積公式得到了該級(jí)數(shù)的兩個(gè)有趣的恒等式,從而推廣了關(guān)于Smarandache-Type可乘函數(shù)的算術(shù)性質(zhì).

Smarandache-Type可乘函數(shù);無(wú)窮級(jí)數(shù);恒等式

1 引言

此外，對(duì)于任意的正整數(shù)n,Smarandache k次冪剩余ak(n)是指滿足nak(n)為一個(gè)完全k次冪的最小正整數(shù).即

從ak(n)的定義中,我們發(fā)現(xiàn)ak(n)仍是一個(gè)可乘函數(shù).設(shè)A表示滿足方程Sk(n)=ak(n)的所有正整數(shù)n的集合.即A={n∈N,Sk(n)=ak(n)}.目前,有許多關(guān)于Smarandache ceil函

其中ζ(s)表示Riemann-zeta函數(shù).

定理2設(shè)k是一個(gè)大于等于2的正整數(shù).則對(duì)于任意的實(shí)數(shù)s＞1,有

2 定理的證明

我們直接給出定理的證明.首先,定義算術(shù)函數(shù)B(n)為

利用同樣的方法,也可以得到

于是完成了定理的證明.

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[2]Ibstedt Surfinig.On the Ocean of Number-a few Smarandache Notions and Similar Topics[M].New Mexico: Erthus University Press,1996.

[3]Sabin Tabirca,Tatiana Tabirca.Some new results concerning the Samarandache ceil function[J].Smarandache notions Journal,2002,13:30-36.

[4]Apostol T M.Introduction to Analytic Number Theory[M].New York:Springer-Verlag,1976.

[5]易媛,亢小玉.關(guān)于Smarandache問(wèn)題研究[M].USA:High American Press,2006.

[6]茍素.關(guān)于Smarandache ceil函數(shù)的一個(gè)方程[J].純粹數(shù)學(xué)與應(yīng)用數(shù)學(xué),2006,22(1):48-50.

Equations on the Smarandache-Type multiplicative function

ZHANG Xiao-beng1,2,TIAN Qing2
(1.Department of Applied Mathematics and Physics,Xi’an University of Post and Telecommunications, Xi’an710121,China;2.Department of Mathematics,Northwest University,Xi’an710127,China)

The main purpose of this paper is using the elementary method and Euler product formula to study the properties of the infinity series involving the Smarandache-Type function,and obtain its two interesting identities.This generalized the properties of Smarandache-Type function.

Smarandache-Type multiplicative function,infinity series,identity

O156.4

A

1008-5513(2009)03-0478-03

2008-09-14.

陜西省教育廳專項(xiàng)科研計(jì)劃項(xiàng)目(08JK437),西安郵電學(xué)院中青年科研基金(105-0449).

張小蹦(1978-),助教,研究方向：數(shù)論及其應(yīng)用.

2000MSC:11B83