Rigorous high-precision computation of the Hurwitz zeta function and its derivatives

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• 作者：Fredrik Johansson
• 关键词：Hurwitz zeta function ; Riemann zeta function ; Arbitrary ; precision arithmetic ; Rigorous numerical evaluation ; Fast polynomial arithmetic ; Power series ; 65D20 ; 68W30 ; 33F05 ; 11 ; 04 ; 11M06 ; 11M35
• 刊名：Numerical Algorithms
• 出版年：2015
• 出版时间：June 2015
• 年：2015
• 卷：69
• 期：2
• 页码：253-270
• 全文大小：1,126 KB
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• 作者单位：Fredrik Johansson (1)
  
  1. RISC, Johannes Kepler University, 4040, Linz, Austria
  
• 刊物类别：Computer Science
• 刊物主题：Numeric Computing
  Algorithms
  Mathematics
  Algebra
  Theory of Computation
  
• 出版者：Springer U.S.
• ISSN：1572-9265

文摘

We study the use of the Euler-Maclaurin formula to numerically evaluate the Hurwitz zeta function ζ(s, a) for \(s, a \in \mathbb {C}\), along with an arbitrary number of derivatives with respect to s, to arbitrary precision with rigorous error bounds. Techniques that lead to a fast implementation are discussed. We present new record computations of Stieltjes constants, Keiper-Li coefficients and the first nontrivial zero of the Riemann zeta function, obtained using an open source implementation of the algorithms described in this paper.