對模n與零同餘之K次根

Abstract

Let k>0 be a positive integer. For each positive n ,let δk(n) be the largest integer such that (δk(n))^k≡0(mod n ).For any positive real number x ,let Dk(x)=∑n<x δk(n) .The purpose of this note is to investigate the order of Dk(x).The main result is the following theorem: Dk(x)/x→ξ(k*1)/ξ(k) for k>2 whereξ(s) is known as Riemann zeta function.