Euler's integral, multiple cosine function and zeta values

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 $\zeta(3)$: $$\zeta(3)=\frac{72\pi^2}{11}\log\left(\frac{3^{\frac1{72}}\mathcal{C}_3\left(\frac16\right)}{\mathcal{C}_2\left(\frac16\right)^{\frac13}}\right).$$