Sums of weighted averages of gcd-sum functions, II

Abstract

We establish two identities involving the Gamma function and Bernoulli polynomials, namely ∑ k ≤ x 1 k s ∑ j = 1 k s log Γ ( j k s ) ∑ d | k , d s | j f ∗ μ ( d ) and ∑ k ≤ x 1 k s ∑ j = 0 k s − 1 B m ∑ d | k , d s | j f ∗ μ ( 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 Dirichlet series associated with the above identities. This paper is a continuation of an earlier work of the authors.