论文标题
Zeta函数的最简单积分及其概括在所有$ \ Mathbb {C} $中有效
Simplest Integrals for the Zeta Function and its Generalizations Valid in All $\mathbb{C}$
论文作者
论文摘要
使用其他方法，我们得出了Riemann Zeta函数及其概括的积分表示（Hurwitz Zeta，$ζ（-K，B）$，polylogarithm，$ \ Mathrm {li} _ { - k} _ { - k}（e^m）$，以及lerch transcendent，$ b，$ ucciinc $ cociince（亚伯 - 普拉纳表情。该方法的略有变化导致不同的公式。我们还介绍了这些功能中的每一个及其部分总和之间的关系。例如，它可以弄清楚$ h _ { - k}（n）$的Taylor系列扩展约为$ n = 0 $（当$ k $是一个正整数时，我们获得了有限的泰勒系列，这只是faulhaber公式）。所使用的方法需要评估$φ\左（e^{2πi\，x}，-2k+1，n+1 \右）+πi\，x \，x \，x \ lest（e^{2πi\，x}，-2k，-2k，n+1 \ right）/k $ x $ x $ 0 $ 0，
Using a different approach, we derive integral representations for the Riemann zeta function and its generalizations (the Hurwitz zeta, $ζ(-k,b)$, the polylogarithm, $\mathrm{Li}_{-k}(e^m)$, and the Lerch transcendent, $Φ(e^m,-k,b)$), that coincide with their Abel-Plana expressions. A slight variation of the approach leads to different formulae. We also present the relations between each of these functions and their partial sums. It allows one to figure, for example, the Taylor series expansion of $H_{-k}(n)$ about $n=0$ (when $k$ is a positive integer, we obtain a finite Taylor series, which is nothing but the Faulhaber formula). The method used requires evaluating the limit of $Φ\left(e^{2πi\,x},-2k+1,n+1\right)+πi\,x\,Φ\left(e^{2πi\,x},-2k,n+1\right)/k$ when $x$ goes to $0$, which in itself already makes for an interesting problem.
