论文标题
Euler的积分，多个余弦函数和Zeta值
Euler's integral, multiple cosine function and zeta values
论文作者
论文摘要
在1769年，欧拉证明了以下结果$$ \ int_0^{\ frac \ pi2} \ log（\sinθ）dθ= - \ frac \ pi2 \ log2。 $$在本文中，作为概括，我们评估了确定的积分$$ \ int_0^xθ^{r-2} \ log \ left（\ cos \ cos \ frac \ theta2 \ right）dθ$$对于$ r = 2,3,4，\ ldots。$。我们表明它可以通过kurokawa和koyama的多个cosine函数$ \ nath C}的特殊值表示，kurokawa和koyama的特殊值是$ \ mathcal的特殊值或和dirichlet lambda功能。 特别是，我们得到以下Zeta值$$的明确表达ζ（3）= \ frac {4π^2} {21} \ log \ left（\ frac {E^{e^{\ frac {4g}π} \ Mathcal {c} _3 _3 \ left（\ frac14 \ right） $ \ MATHCAL {C} _3 \ left（\ frac14 \ right）$是kurokawa和Koyama的多余余弦函数$ \ Mathcal {c} _3（x）$ at $ \ frac14 $的特殊值。此外，我们通过zeta函数，$ l $ functions或polyrogarithms证明了多个余弦函数对数的几个串联表示。其中一个导致了另一种表达 $ζ（3）$： $ζ（3）= \ frac {72π^2} {11}} \ log \ left（\ frac {3^{\ frac1 {72}}} \ Mathcal {c} _3 \ left （\ frac16 \ right）}} {\ mathcal {c} _2 \ left（\ frac16 \ right）^{\ frac13}}} \ right）。$$
In 1769, Euler proved the following result $$ \int_0^{\frac\pi2}\log(\sin θ) dθ=-\frac\pi2 \log2. $$ In this paper, as a generalization, we evaluate the definite integrals $$ \int_0^x θ^{r-2}\log\left(\cos\frac\theta2\right)dθ$$ 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 $$ ζ(3)=\frac{4π^2}{21}\log\left(\frac{e^{\frac{4G}π}\mathcal{C}_3\left(\frac14\right)^{16}}{\sqrt2}\right), $$ 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 $ζ(3)$: $$ζ(3)=\frac{72π^2}{11}\log\left(\frac{3^{\frac1{72}}\mathcal{C}_3\left(\frac16\right)}{\mathcal{C}_2\left(\frac16\right)^{\frac13}}\right).$$
