Lambert series of logarithm, the derivative of Deninger's function $R(z)$ and a mean value theorem for $\zeta \left(\frac{1}{2}-it\right){\zeta }^{\prime }\left(\frac{1}{2}+it\right)$

Abstract

An explicit transformation for the series $\sum _{n=1}^{\infty }\frac{\log (n)}{e^{ny}-1},$ 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 $\sum _{n=1}^{\infty }\frac{\log (n)}{e^{ny}-1}$ as $y\to 0$. An application of the latter is that it gives the asymptotic expansion of ${\int }_0^{\infty }\zeta \left(\frac{1}{2}-it\right){\zeta }^{\prime }\left(\frac{1}{2}+it\right)e^{-\delta t}dt$ as $\delta \to 0$.

25 pages, submitted for publication