  Let $\psi$ denote the Dedekind totient function defined by $
\psi(n)=\sum_{d|n}d\mu^2\l({n}/{d}\r) $ with $\mu$ being the M\"{o}bius
function. We shall consider the $k$-th Riesz mean of the arithmetical function
$n/\psi(n)$ for any non-negative integer $k$ on the assumptions that the
Riemann Hypothesis is true, and all the zeros $\rho$ on the critical line of
the Riemann zeta function $\zeta$ are simple. Our result is an explicit
representation of the error term in the formula obtained in a previous work of
the second author and I. Kiuchi \cite{IK}. We also give an improvement on the
error estimate under the assumption of the Gonek-Hejhal Hypothesis. And, we
propose a proposition that is equivalent to the Riemann Hypothesis.
