Euler's integral, multiple cosine function and zeta values

TL;DR

This paper generalizes Euler's integral involving logarithms and trigonometric functions, expressing zeta values and integrals through multiple cosine functions and special functions, providing new series representations and explicit formulas.

Contribution

It introduces new integral evaluations related to multiple cosine functions and derives explicit formulas for zeta values using these special functions.

Findings

Explicit evaluation of integrals involving logarithms and cosine functions.

New formulas expressing zeta(3) in terms of multiple cosine functions.

Series representations of logarithms of multiple cosine functions using zeta and polylogarithm functions.

Abstract

In 1769, Euler proved the following result $$ \int_0^{\frac\pi2}\log(\sin \theta) d\theta=-\frac\pi2 \log2. $$ In this paper, as a generalization, we evaluate the definite integrals $$ \int_0^x \theta^{r-2}\log\left(\cos\frac\theta2\right)d\theta $$ for $r=2,3,4,\ldots.$ We show that it can be expressed by the special values of Kurokawa and Koyama's multiple cosine functions $\mathcal{C}_r(x)$ or by the special values of alternating zeta and Dirichlet lambda functions. In particular, we get the following explicit expression of the zeta value $$ \zeta(3)=\frac{4\pi^2}{21}\log\left(\frac{e^{\frac{4G}{\pi}}\mathcal{C}_3\left(\frac14\right)^{16}}{\sqrt2}\right), $$ where $G$ is Catalan's constant and $\mathcal{C}_3\left(\frac14\right)$ is the special value of Kurokawa and Koyama's multiple cosine function $\mathcal{C}_3(x)$ at $\frac14$. Furthermore, we prove several series representations for the logarithm of multiple cosine functions $\log\mathcal{C}_r\left(\frac x{2}\right)$ by zeta functions, $L$-functions or polylogarithms. One of them leads to another expression of $\zeta(3)$: $$\zeta(3)=\frac{72\pi^2}{11}\log\left(\frac{3^{\frac1{72}}\mathcal{C}_3\left(\frac16\right)}{\mathcal{C}_2\left(\frac16\right)^{\frac13}}\right).$$