Relating log-tangent integrals with the Riemann zeta function

Abstract

We show that integrals involving log-tangent function, with respect to certain square-integrable functions on $(0, \pi/2)$, can be evaluated by some series involving the harmonic number. Then we use this result to establish many closed forms relating to the Riemann zeta function at odd positive integers. In addition, we show that the log-tangent integral with respect to the Hurwitz zeta function defines a meromorphic function and that its values depend on the Dirichlet series $\zeta_h(s) :=\sum_{n = 1}^\infty h_n n^{-s}$, where $h_n = \sum_{k=1}^n(2k-1)^{-1}$.