Euler’s integral, multiple cosine function and zeta values

Abstract

Abstract In 1769, Euler proved the following result: ∫ 0 π 2 log ⁡ ( sin ⁡ θ ) ⁢ 𝑑 θ = - π 2 ⁢ log ⁡ 2 . \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 ∫ 0 x θ r - 2 ⁢ log ⁡ ( cos ⁡ θ 2 ) ⁢ 𝑑 θ \int_{0}^{x}\theta^{r-2}\log\biggl{(}\cos\frac{\theta}{2}\biggr{)}\,d\theta for r = 2 , 3 , 4 , … r=2,3,4,\dots  . We show that it can be expressed by the special values of Kurokawa and Koyama’s multiple cosine functions 𝒞 r ⁢ ( x ) {\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: ζ ⁢ ( 3 ) = 4 ⁢ π 2 21 ⁢ log ⁡ ( e 4 ⁢ G π ⁢ 𝒞 3 ⁢ ( 1 4 ) 16 2 ) , \zeta(3)=\frac{4\pi^{2}}{21}\log\Biggl{(}\frac{e^{\frac{4G}{\pi}}\mathcal{C}_{% 3}\bigl{(}\frac{1}{4}\bigr{)}^{16}}{\sqrt{2}}\Biggr{)}, where G is Catalan’s constant and 𝒞 3 ⁢ ( 1 4 ) {\mathcal{C}_{3}(\frac{1}{4})} is the special value of Kurokawa and Koyama’s multiple cosine function 𝒞 3 ⁢ ( x ) {\mathcal{C}_{3}(x)} at 1 4 {\frac{1}{4}} . Furthermore, we prove several series representations for the logarithm of multiple cosine functions log ⁡ 𝒞 r ⁢ ( x 2 ) {\log\mathcal{C}_{r}(\frac{x}{2})} by zeta functions, L-functions or polylogarithms. One of them leads to another expression of ζ ⁢ ( 3 ) {\zeta(3)} : ζ ⁢ ( 3 ) = 72 ⁢ π 2 11 ⁢ log ⁡ ( 3 1 72 ⁢ 𝒞 3 ⁢ ( 1 6 ) 𝒞 2 ⁢ ( 1 6 ) 1 3 ) . \zeta(3)=\frac{72\pi^{2}}{11}\log\Biggl{(}\frac{3^{\frac{1}{72}}\mathcal{C}_{3% }\bigl{(}\frac{1}{6}\bigr{)}}{\mathcal{C}_{2}\bigl{(}\frac{1}{6}\bigr{)}^{% \frac{1}{3}}}\Biggr{)}.