Sums of weighted averages of gcd-sum functions II

TL;DR

This paper derives new identities involving the Gamma function and Bernoulli polynomials related to gcd-sum functions, providing asymptotic formulas and Dirichlet series analysis for various multiplicative functions.

Contribution

It introduces novel identities for gcd-sum functions involving Gamma and Bernoulli polynomials, extending previous work with asymptotic and Dirichlet series results.

Findings

Established identities involving Gamma function and Bernoulli polynomials

Provided asymptotic formulas for identities with multiplicative functions

Analyzed Dirichlet series associated with the identities

Abstract

In this paper, we establish the following two identities involving the Gamma function and Bernoulli polynomials, namely $$ \sum_{k\leq x}\frac{1}{k^s} \sum_{j=1}^{k^s}\log\Gamma\left(\frac{j}{k^s}\right) \sum_{\substack{d|k \\ d^{s}|j}}f*\mu(d) \quad {\rm and } \quad \sum_{k\leq x}\frac{1}{k^s}\sum_{j=0}^{k^{s}-1} B_{m}\sum_{\substack{d|k \\ d^{s}|j}} f*\mu(d) $$ with any fixed integer $s> 1$ and any arithmetical function $f$. We give asymptotic formulas for them with various multiplicative functions $f$. We also consider several formulas of the Dirichlet series associated with the above identities. This paper is a continuation of an earlier work of the authors.