Relating log-tangent integrals with the Riemann zeta function

TL;DR

This paper establishes new connections between log-tangent integrals, harmonic series, and the Riemann zeta function at odd integers, providing closed-form expressions and exploring their meromorphic properties.

Contribution

It introduces novel evaluations of log-tangent integrals in terms of harmonic numbers and relates these to the Riemann zeta function at odd integers, expanding understanding of their interrelations.

Findings

Closed-form expressions for log-tangent integrals involving harmonic numbers.

New relations between log-tangent integrals and zeta function at odd integers.

Log-tangent integrals define meromorphic functions depending on Dirichlet series.

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 $h_n = \sum_{k=1}^n(2k-1)^{-1}$.