Lambert series of logarithm, the derivative of deninger's function R(z) and a mean value theorem for ζ(12−it) ζ′(12+it)

Abstract

An explicit transformation for the series ∑n=1∞log(n)eny−1, Re(y)>0, which takes y to 1/y, is obtained for the first time. This series transforms into a series containing ψ1(z), the derivative of Deninger's function R(z). In the course of obtaining the transformation, new important properties of ψ1(z) are derived, as is a new representation for the second derivative of the two-variable Mittag-Leffler function E2,b(z) evaluated at b=1. Our transformation readily gives the complete asymptotic expansion of ∑n=1∞log(n)eny-1 as y→0. An application of the latter is that it gives the asymptotic expansion of ∫∞0ζ(12-it)ζ′(12+it)e-δtdt as δ→0.