Infinite sum of the product of exponential and logarithmic functions, its analytic continuation, and application (Q2654534)

The sum in the title is \(S_1(x)= \sum^\infty_{k=1} \exp(-2\pi kx)\log k\), defined originally for \(x> 0\), and then extended analytically to complex values of \(x\). Explicit formulas express \(S_1(x)\) as an elementary function of \(x\) plus two power series in \(x\) whose coefficients involve the Riemann zeta function and its derivatives evaluated at positive integers. Applications include evaluation of special functions of Ramanujan at rational values.