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.
Categories: math.NT