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

Abstract

We show that the function S_(1)(x) = ∑_(k=1)^∞ e^(-2πkx) log k can be expressed as the sum of a simple function and an infinite series, whose coefficients are related to the Riemann zeta function. Analytic continuation to the imaginary argument S_(1)(ix) = K_0(x) is made. For x = p/q where p and q are integers with p < q, closed finite sum expressions for K_0(p/q) and K_1(p/q) are derived. The latter results enable us to evaluate Ramanujan's function ψ(x) = ∑_(k=1)^∞ [(logk)/k - (log(k+x))/(k+x)] for x = -2/3, -3/4, and -5/6, confirming what Ramanujan claimed but did not explicitly reveal in his Notebooks. The interpretation of a pair of apparently inscrutable divergent series in the notebooks is discussed. They reveal hitherto unsuspected connections between Ramanujan's ψ(x), K_0(x), K_1(x), and the classical formulas of Gauss and Kummer for the digamma function