伪Smarandache无平方因子函数与D(n)函数的混合均值

The Hybrid Mean of the Pseudo Smarandache Squarefree Function and D(n)Function

伪Smarandache无平方因子函数Z_w(n)的定义为有最小的正整数m使得n|m^n,即有Z_w(n)=min{m:n|m^n,m∈N}.而数论函数D(n)的定义为存在最小正整数m使得n|d(1)d(2)d(3)…d(m)(d(n)为Dirichlet除数函数),即D(n)=min{m:n|d(i),m∈N}.本文利用初等和解析方法研究这两个函数的混合均值问题,并给出其两个渐近公式.同时通过前人的结论提出猜想,最后推广了定理2的结论.

For any positive integer n,the Pseudo Smarandache Squarefree function Z_w(n)is defined as the smallest positive integer m such that n|m^n,that is Z_w(n) = min{m:n|m^n,m ∈ N}.And the number theory function D(n) is defined as the smallest positive integer m such that n divide product n|d(1)d(2)d(3)…d(m)(d(n),where d(n) is the famous Dirichlet divisor function,that is D(n)= min{m:n|d(i),m∈N}.The main purpose of this paper is to use the elementary and analytic methods to study the hybrid mean value properties,and obtain asymptotic formula for them.At the same time,through the conclusions of the predecessors and the conjectures,the conclusion of Theorem 2 is finally extended.