Sums of gcd-sum functions with weights concerning the Gamma function and Bernoulli polynomials

Isao Kiuchi, Sumaia Saad Eddin

Published 2018-01-11Version 1

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 the above with various multiplicative functions $f$. We also consider several formulas of Dirichlet series having coefficients $\gcd$-sum functions with weights concerning the Gamma function and Bernoulli polynomials.