Euler's integral, multiple cosine function and zeta values

Abstract

In 1769, Euler proved the following result $${\int }_0^{\frac{\pi }{2}}\log (\sin \theta )d\theta =-\frac{\pi }{2}\log 2.$$ In this paper, as a generalization, we evaluate the definite integrals $${\int }_0^x{\theta }^{r-2}\log \left(\cos \frac{\theta }{2}\right)d\theta$$ for $r=2,3,4,\dots .$ 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 $$\zeta (3)=\frac{4{\pi }^2}{21}\log \left(\frac{e^{\frac{4G}{\pi }}{\mathcal{C}}_3{\left(\frac{1}{4}\right)}^{16}}{\sqrt{2}}\right),$$ where $G$ is Catalan's constant and ${\mathcal{C}}_3\left(\frac{1}{4}\right)$ is the special value of Kurokawa and Koyama's multiple cosine function ${\mathcal{C}}_3(x)$ at $\frac{1}{4}$. 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 $\zeta (3)$: $$\zeta (3)=\frac{72{\pi }^2}{11}\log \left(\frac{3^{\frac{1}{72}}{\mathcal{C}}_3\left(\frac{1}{6}\right)}{{\mathcal{C}}_2{\left(\frac{1}{6}\right)}^{\frac{1}{3}}}\right).$$

30 pages. We are grateful to Professor Jean-Paul Allouche for his interested in this paper and for his many helpful comments and suggestions