Riemann, Hurwitz and Hurwitz-Lerch Zeta Functions and Associated Series and Integrals
Riemann, Hurwitz and Hurwitz-Lerch Zeta Functions and Associated Series and Integrals
复制标题
Riemann、Hurwitz 和 Hurwitz-Lerch Zeta 函数以及相关级数和积分
DOI：
--
复制
发表时间：
2012
期刊：
影响因子：
--
通讯作者：
H. Srivastava
中科院分区：
文献类型：
--
作者：
H. Srivastava
本文的主要目的是对Riemann Zeta函数（zeta（s））、Hurwitz（或广义）Zeta函数（zeta（s，a））和Hurwitz-Lerch Zeta函数（Phi（z，s，a））及其各种有趣的扩展和推广的一些最新发展作一综述。我们首先研究与（zeta left（s））的求值和表示相关的问题
right））when（s in mathbb{N} setminus left {1
ight }），（mathbb{N}）是自然数的集合，强调了（zeta left（2n + 1）的几类有趣的快速收敛级数表示
right））（left（n in mathbb{N}
（八）近年来发展起来的。在这里考虑的许多计算上有用的特殊情况中的两种中，观察到（zeta left（3
right））可以用比欧拉著名公式收敛得快得多的级数来表示，也可以用最近罗杰·阿维尼翁（1916-1994）在证明（zeta left（3））的无理性时所用的级数来表示
（八））。在Linux上使用Mathematica（4.0版）进行的符号和数值计算表明，其中一个级数只有50项能够产生小数点后7位的精度。我们还考虑了与Hurwitz-Lerch Zeta函数（Phi（z，s，a））相关的各种级数和积分以及它的各种有趣的扩展和推广。
The main object of this article is to present a survey-cum-expository account of some recent developments involving the Riemann Zeta function (zeta (s)), the Hurwitz (or generalized) Zeta function (zeta (s,a)), and the Hurwitz-Lerch Zeta function (Phi (z,s,a)) as well as its various interesting extensions and generalizations. We first investigate the problems associated with the evaluations and representations of (zeta left (s
ight )) when (s in mathbb{N} setminus left {1
ight }), (mathbb{N}) being the set of natural numbers, emphasizing upon several interesting classes of rapidly convergent series representations for (zeta left (2n + 1
ight )) (left (n in mathbb{N}
ight )) which have been developed in recent years. In two of many computationally useful special cases considered here, it is observed that (zeta left (3
ight )) can be represented by means of series which converge much more rapidly than that in Euler’s celebrated formula as well as the series which was used more recently by Roger Apery (1916–1994) in his proof of the irrationality of (zeta left (3
ight )). Symbolic and numerical computations using Mathematica (Version 4.0) for Linux show, among other things, that only 50 terms of one of these series are capable of producing an accuracy of seven decimal places. We also consider a variety of series and integrals associated with the Hurwitz-Lerch Zeta function (Phi (z,s,a)) as well as its various interesting extensions and generalizations.