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

TL;DR

This paper derives a new explicit transformation for a logarithmic series involving exponential terms, linking it to Deninger's function and Mittag-Leffler functions, with applications to asymptotic analysis of zeta function products.

Contribution

It introduces a novel transformation for a logarithmic series, connecting it to derivatives of Deninger's function and providing new properties and representations for special functions.

Findings

Derived an explicit y to 1/y transformation for the series.

Obtained the asymptotic expansion of the series as y approaches 0.

Applied the transformation to analyze the asymptotic behavior of zeta function products.

Abstract

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 $\delta\to0$.