Lambert series of logarithm, the derivative of Deninger’s function R(z), and a mean value theorem for ζ (21 - it) ζ′ (21 + it)

An explicit transformation for the series (Formula presented), or equivalently, (Formula presented) ∞ for Re(y) > 0, which takes y to 1/y, is obtained for the first time. This series transforms into a series containing the derivative of R(z), a function studied by Christopher Deninger while obtaining an analog of the famous Chowla–Selberg formula for real quadratic fields. In the course of obtaining the transformation, new important properties of ψ1(z) (the derivative of R(z)) are needed as is a new representation for the second derivative of the two-variable Mittag-Leffler function E<inf>2,b</inf>(z) evaluated at b = 1, all of which may seem quite unexpected at first glance. Our transformation readily gives the complete asymptotic expansion of (Formula presented) as y → 0 which was also not known before. An application of the latter is that it gives the asymptotic expansion of (Formula presented).