Sums of averages of gcd-sum functions

JOURNAL OF NUMBER THEORY

卷:176

Article

Kiuchi, Isao¹ 

[1] Yamaguchi Univ, Fac Sci, Dept Math Sci, Yoshida 1677-1, Yamaguchi 7538512, Japan

关键词: gcd-sum functions;    Euler totient function;    Dedekind function;    Anderson-Apostol sums;    Dirichlet divisor problem;    Asymptotic results on arithmetical functions;   

DOI  :  10.1016/j.jnt.2016.12.021

来源: Elsevier

【 摘 要 】

Let gcd (k, j) be the greatest common divisor of the integers k and j. We establish some asymptotic formulas for weighted averages of the gcd-sum functions, that is Sigma(k <= x) 1/k(r+1) Sigma(k)(j=1) j(T) f (gcd(k, j)) with f = id, phi,phi(s), psi and psi(s) for any fixed positive integers r and s, where phi,phi(s), psi and psi(s) and are the Euler, the Jordan, the Dedekind and the generalized Dedekind function, respectively, and also prove the mean square formulas of the gcd-sum function Sigma(k <= x) 1/k(r+1). Sigma(k)(j=1) j(r) phi(gcd(k,j)) and Sigma(k <= x) 1/k(r+1) Sigma(k)(j=1) j(r) psi(gcd(k,j)). (C) 2017 Published by Elsevier Inc.

【 授权许可】

Free