An explicit transformation for the series \sum \limits _{n=1}^{\infty }\displaystyle \frac {\log (n)}{e^{ny}-1}$\sum \limits _{n=1}^{\infty }\displaystyle \frac {\log (n)}{e^{ny}-1}$, or equivalently, \sum \limits _{n=1}^{\infty }d(n)\log (n)e^{-ny}$\sum \limits _{n=1}^{\infty }d(n)\log (n)e^{-ny}$ for Re(y)>0$(y)>0$, which takes y to 1/y$1/y$, is obtained for the first time. This series transforms into a series containing the derivative of R(z)$R(z)$, a function studied by Christopher Deninger while obtaining an analog of the famous Chowla–Selberg formula for real quadratic fields. In the course of obtaining the transformation, new important properties of \psi _1(z)$\psi _1(z)$ (the derivative of R(z)$R(z)$) are needed as is a new representation for the second derivative of the two-variable Mittag-Leffler function E_{2, b}(z)$E_{2, b}(z)$ evaluated at b=1$b=1$, all of which may seem quite unexpected at first glance. Our transformation readily gives the complete asymptotic expansion of \sum \limits _{n=1}^{\infty }\displaystyle \frac {\log (n)}{e^{ny}-1}$\sum \limits _{n=1}^{\infty }\displaystyle \frac {\log (n)}{e^{ny}-1}$ as y\to 0$y\to 0$ which was also not known before. 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$\displaystyle \int _{0}^{\infty }\zeta \left (\frac {1}{2}-it\right )\zeta '\left (\frac {1}{2}+it\right )e^{-\delta t}\, dt$ as \delta \to 0$\delta \to 0$.