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

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 ?1(z), the derivative of Deninger's function R(z). In the course of obtaining the transformation, new important properties of ?1(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 ??0?(12-it)??(12+it)e-?tdt as ??0.