INFINITE SUM OF THE PRODUCT OF EXPONENTIAL AND LOGARITHMIC FUNCTIONS, ITS ANALYTIC CONTINUATION, AND APPLICATION

摘要

We show that the function S_1(x) = ∞/∑/k=1 e~(-2πkx) log k can be ex-rnpressed 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)-iK_1(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 φ(x) = ∞/∑/k=1((log k)/k-(log(k+x)/(k+x))) for x=-2/3,-3/4 and -5/6, confirming whatrnRamanujan 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 φ(x), K_0(x), K_1(x), and the classical formulas of Gauss and Kummer for the digamma function.