On sums of weighted averages of gcd-sum functions

2018/01/11 by Isao Kiuchi, Kiuchi, Isao, Sumaia Saad Eddin +1
Mathematics · #11A25 #11N37 #11Y60 #Advanced Mathematical Identities #Advanced Mathematical Theories #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.1801.03647

openalex publication_date 2018/01/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01

Abstract

Let gcd(j,k) be the greatest common divisor of the integers j and k. In this paper, we give several interesting asymptotic formulas for weighted averages of the gcd-sum function f(gcd(j,k)) and the function ∑_{d|k, d^s|j}(f*μ)(d) for any positive integers j and k, namely ∑_{k≤ x}\frac1k^r+1∑_j=1^kj^rf(gcd(j,k)) and ∑_{k≤ x}\frac1k^s(r+1)∑_j=1^ksj^r ∑_\substackd|k d^s|j(f*μ)(d), with any fixed integer s> 1 and any arithmetical function f. We also establish mean value formulas for the error terms of asymptotic formulas for partial sums of gcd-sum functions f(gcd(j,k)).

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