Lambert series of logarithm and a mean value theorem for $\zeta(\frac{1}{2}-it)\zeta'(\frac{1}{2}+it)$

An explicit transformation for the series $\sum_{n=1}^{\infty}d(n)\log(n)e^{-ny},$ Re$(y)>0$, which takes $y$ to~$\frac1y$, is obtained. This series transforms into a series containing $\psi_1(z)$, the derivative of~$R(z)$. The latter is a function studied by Christopher Deninger while obtaining an analogue of the famous Chowla--Selberg formula for real quadratic fields. 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 $\sum_{n=1}^{\infty}d(n)\log(n)e^{-ny}$ as $y\to0$. This, in turn, gives the asymptotic expansion of $\int_{0}^{\infty}\zeta\left(\frac{1}{2}-it\right)\zeta'\left(\frac{1}{2}+it\right)e^{-\delta t}\, dt$ as $\delta\to0$. This is joint work with Soumyarup Banerjee and Shivajee Gupta.