On an intriguing integral and some series related to $\zeta(4)$

Abstract

An intriguing log-cosine integral is fully analyzed and shown to have value a rational multiple of 7r(4), C being the Riemann zeta function. From this we deduce by means of generating functions and Parseval's identity the sums of certain series previously established by a completely different method. \ -(I + -+ + -)smnt; *-^ nv 2 n' n=\ and (ii) numerically determine the value of the desired sum via Parseval's equation. This led to equation (12) below and an evaluation of the sum to 25 digits