  An explicit transformation for the series
$\sum\limits_{n=1}^{\infty}\displaystyle\frac{\log(n)}{e^{ny}-1},$ Re$(y)>0$,
which takes $y$ to $1/y$, is obtained for the first time. This series
transforms into a series containing $\psi_1(z)$, the derivative of Deninger's
function $R(z)$. In the course of obtaining the transformation, new important
properties of $\psi_1(z)$ are derived, as is a new representation for the
second derivative of the two-variable Mittag-Leffler function $E_{2, b}(z)$
evaluated at $b=1$. Our transformation readily gives the complete asymptotic
expansion of $\sum\limits_{n=1}^{\infty}\displaystyle\frac{\log(n)}{e^{ny}-1}$
as $y\to0$. An application of the latter is that it gives the asymptotic
expansion of $
\displaystyle\int_{0}^{\infty}\zeta\left(\frac{1}{2}-it\right)\zeta'\left(\frac{1}{2}+it\right)e^{-\delta
t}\, dt$ as $\delta\to0$.
