Sums of weighted averages of gcd-sum functions, II

Abstract

We establish two identities involving the Gamma function and Bernoulli polynomials, namely

$$\sum _{k\le x}\frac{1}{k^s}\sum _{j=1}^{k^s}log\mathrm{\Gamma }\left(\frac{j}{k^s}\right)\sum _{d|k,d^s|j}f\ast \mu \left(d\right)\text{ and}\sum _{k\le x}\frac{1}{k^s}\sum _{j=0}^{k^s-1}{B}_m\sum _{d|k,d^s|j}f\ast \mu \left(d\right)$$

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 Dirichlet series associated with the above identities. This paper is a continuation of an earlier work of the authors.