  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.
