Computational Strategies for the Riemann Zeta Function

Computational Strategies for the Riemann Zeta Function (2000)

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by Jonathan M. Borwein , David M. Bradley , Richard E. Crandall

Venue:	Journal of Computational and Applied Mathematics
Citations:	34 - 8 self

BibTeX

@ARTICLE{Borwein00computationalstrategies,
    author = {Jonathan M. Borwein and David M. Bradley and Richard E. Crandall},
    title = {Computational Strategies for the Riemann Zeta Function},
    journal = {Journal of Computational and Applied Mathematics},
    year = {2000},
    volume = {121},
    pages = {247--296}
}

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Abstract

We provide a compendium of evaluation methods for the Riemann zeta function, presenting formulae ranging from historical attempts to recently found convergent series to curious oddities old and new. We concentrate primarily on practical computational issues, such issues depending on the domain of the argument, the desired speed of computation, and the incidence of what we call "value recycling".

Citations

644	A Course of Modern Analysis - Whittaker, Watson - 1996
294	Introduction to Analytic Number Theory - Apostol - 1976
208	Polylogarithms and Associated Functions - Lewin - 1981
157	Riemann’s zeta function - Edwards - 2001
138	Ramanujan’s notebooks. Part V - Berndt - 1998
131	The Theory of the Riemann Zeta-Function - Titchmarsh - 1951
119	Prime Numbers - A Computational Perspective - Crandall, Pomerance - 2000
113	Fast Fourier transforms for nonequispaced data - Dutt, Rokhlin - 1993
108	The Riemann zeta-function - Ivić - 1985
105	The New Book of Prime Number Records - Ribenboim - 1995
85	On the rapid computation of various polylogarithmic constants - Bailey, Borwein, et al. - 1997
73	Fast multiple-precision evaluation of elementary functions,” Journal of the association for computing machinery - Brent - 1976
72	Trace formula in non-commutative Geometry and the zeros of the Riemann zeta function - Connes - 1999
67	On the distribution of spacings between the zeros of the zeta function - Odlyzko - 1987
66	Applied and - Henrici - 1974
64	Prime Numbers and Computer Methods for Factorization - Riesel - 1994
59	der Poorten, A proof that Euler missed ... Apéry’s proof of the irrationality of ζ(3), Mathematics Intelligencer 1 - van - 1978
59	A course of modern analysis, 4th edition - Whittaker, Watson - 1927
46	Spectral theory of the Riemann zeta-function - Motohashi - 1997
43	Fast algorithms for multiple evaluations of the Riemann zeta function - Odlyzko, Schnhage
42	On the random character of fundamental constant expansions - Bailey, Crandall
32	Sur la distribution des nombres premiers - Littlewood - 1914
30	Evidence for a spectral interpretation of the zeros of L-functions - Rubinstein - 1998
26	te Riele, Disproof of the Mertens conjecture - Odlyzko, J - 1985
26	Ramanujan’s notebooks. Part II - Berndt - 1989
24	Topics in Advanced Scientific Computation - Crandall - 1996
23	On the zeros of the Riemann zeta function in the critical strip - Lune, Diele, et al. - 1986
22	Hypergeometric series acceleration via the WZ method’, The Wilf Festschrift - Amdeberhan, Zeilberger - 1996
21	Fast Algorithms for Polynomial Interpolation, Integration and Differentiation - Dutt, Gu, et al. - 1996
21	Fast multiprecision evaluation of series of rational numbers - Haible, Papanikolaou - 1997
21	Strassen: “Schnelle Multiplikation grosser Zahlen”. Computing 7 - Schönhage, V - 1971
20	Polylogarithmic ladders, hypergeometric series and the ten millionth digits of ζ(3) and ζ(5 - Broadhurst - 1998
18	On the evaluation of Euler sums - Crandall, Buhler - 1995
17	Lagarias, “Complements to Li’s criterion for the Riemann hypothesis - Bombieri, C - 2004
17	Sums of products of Bernoulli numbers - Dilcher
17	The positivity of a sequence of numbers and the Riemann hypothesis - Li - 1997
16	The Riemann hypothesis - Borwein, Choi, et al. - 2007
13	On the zeros of the Riemann zeta function in the critical strip - Brent - 1979
13	An Efficient Algorithm for the Riemann Zeta Function - Borwein - 1995
12	Neue Herleitung und explizite Restabschätzung der Riemann-Siegel-Formel - Gabcke - 1979
12	The Riemann zeta-function, Walter de Gruyter - Karacuba, Voronin - 1992
12	The distribution of prime numbers, Cambridge Tracts - Ingham - 1990
11	Irregular primes to one million - Butler, Crandall, et al. - 1992
10	Zeta function expansions of classical constants, unpublished manuscript; available at http://algo.inria.fr/flajolet/Publications/landau.ps - Flajolet, Vardi
9	On the Khintchine constant - Bailey, Borwein, et al. - 1997
9	The 10 -th Zero of the Riemann Zeta Function and 175 Million of its Neighbors - Odlyzko - 1992
9	Quantum chaology - Berry - 1987
8	Some series of the zeta and related functions, Analysis - Adamchik, Srivastava - 1998
8	The complexity of number-theoretic constants - Bach - 1997
8	An asymptotic representation for the Riemann zeta function on the critical line - Paris - 1994