Sums of Averages of GCD-Sum Functions II

Abstract

Let gcd (k,j) denote the greatest common divisor of the integers k and j, and let r be any fixed positive integer. Define Mr(x;f):=∑k≤x1kr+1∑j=1kjrf(gcd(j,k))\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} M_r(x; f) := \\sum _{k\\le x}\\frac{1}{k^{r+1}}\\sum _{j=1}^{k}j^{r}f(\\gcd (j,k)) \\end{aligned}$$\\end{document}for any large real number xge 5, where f is any arithmetical function. Let phi , and psi denote the Euler totient and the Dedekind function, respectively. In this paper, we refine asymptotic expansions of M_r(x; mathrm{id}), M_r(x;{phi }) and M_r(x;{psi }). Furthermore, under the Riemann Hypothesis and the simplicity of zeros of the Riemann zeta-function, we establish the asymptotic formula of M_r(x;mathrm{id}) for any large positive number x>5 satisfying x=[x]+frac{1}{2}.