Sums of weighted averages of gcd-sum functions II

Abstract

In this paper, we establish the following two identities involving the Gamma function and Bernoulli polynomials, namely $$\sum _{k\le x}\frac{1}{k^s}\sum _{j=1}^{k^s}\log \Gamma \left(\frac{j}{k^s}\right)\sum _{\begin{array}{c}d|k\\ d^s|j\end{array}}f\ast \mu (d)\mathrm{and}\sum _{k\le x}\frac{1}{k^s}\sum _{j=0}^{k^s-1}{B}_m\sum _{\begin{array}{c}d|k\\ d^s|j\end{array}}f\ast \mu (d)$$ 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.

arXiv comment17 pages