合数模上Hardy和的均值  

The Mean Values of Hardy Sum with Composite Moduli

作　　者：岳霞霞 YUE Xia-xia(Department of Basic Courses,Shanxi Institute of Technology,Yangquan 045000,Shanxi,China)

机构地区：[1]山西工程技术学院基础部,山西阳泉045000

出　　处：《山西师范大学学报（自然科学版）》2020年第3期4-9,共6页Journal of Shanxi Normal University(Natural Science Edition)

摘　　要：设q为奇数且q≥5,h为任一整数,本文研究如下形式的Hardy和[1]H(h,q)=Σj=1^q-1(-1)j+1+[hj/q],其中[x]表示不超过x的最大整数.Hardy和的均值定理一直是数论研究的重要内容之一.本文运用特征和的Fourier展式、Dirichlet L-函数的均值定理、可乘函数的性质以及Euler乘积公式,对合数模上Hardy和的均值做了进一步研究,并得到如下结论.Σa≤q/3Σb≤q/3H(2ab,q)=1/5q^2Πpα‖q(1-1/p^2)(1-1/p^3α-(1+1/p+1/p^2)(1/p2α-1/p^3α))/(1+1/p^2)(1+1/p+1/p^2)+O(q^1+∈)其中Πpα‖q表示对同时满足p^α|q与p^α+1■q的所有素数p求积,∈为任意小的正数.Under the condition that q,an odd number,is no smaller than 5,and h is an integer,this paper studies the Hardy Sum,which is defined asH(h,q)=Σj=1^q-1(-1)^j+1+[hj/q],where[x]means the integer that is no larger than x.Because the Mean Value Theorem of the Hardy Sum has always been one of the most significant parts in Studies in the Theory of Numbers,this paper further studies the mean value of the Hardy Sum in short interval by using the Fourier expansion of characteristic sum,the mean value theorems of Dirichlet L-functions,the properties of multiplicative function and Euler product formula.And the result is as following:Σa≤q/3Σb≤q/3H(2ab,q)=1/5q^2Πpα‖q(1-1/p^2)(1-1/p^3α-(1+1/p+1/p^2)(1/p2α-1/p^3α))/(1+1/p^2)(1+1/p+1/p^2)+O(q^1+∈)in which Πpα‖q denotes the product of all the prime number p that matches the condition p^α|q and p^α+1■q,and ∈ isan limitless-small positive number.

关 键 词：Hardy和 DIRICHLET级数 Abel恒等式 渐近公式 

分 类 号：O156.4[理学—数学]