Evaluation of Log-tangent Integrals by series involving $\zeta(2n+1)$

TL;DR

This paper demonstrates that integrals of the log-tangent function against square-integrable functions on [0, π/2] can be expressed using series involving the Riemann Zeta function at odd positive integers, providing a new analytical approach.

Contribution

It introduces a method to evaluate log-tangent integrals through series involving ζ(2n+1), linking special functions with the Riemann Zeta function.

Findings

Integrals of log-tangent functions can be expressed via series involving ζ(2n+1).

The approach applies to any square-integrable function on [0, π/2].

Provides a new analytical tool for evaluating such integrals.

Abstract

In this note, we show that the values of integrals of the log-tangent function with respect to any square-integrable function on $\left[0 , \frac{\pi}{2} \right]$ may be determined by a finite or infinite sum involving the Riemann Zeta-function at odd positive integers.