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Computational strategies for the Riemann zeta function
, 2000
"... 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 th ..."
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Cited by 45 (9 self)
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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”.
The Incomplete Gamma Functions Since Tricomi
 In Tricomi's Ideas and Contemporary Applied Mathematics, Atti dei Convegni Lincei, n. 147, Accademia Nazionale dei Lincei
, 1998
"... The theory of the incomplete gamma functions, as part of the theory of conuent hypergeometric functions, has received its rst systematic exposition by Tricomi in the early 1950s. His own contributions, as well as further advances made thereafter, are surveyed here with particular emphasis on asy ..."
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Cited by 14 (1 self)
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The theory of the incomplete gamma functions, as part of the theory of conuent hypergeometric functions, has received its rst systematic exposition by Tricomi in the early 1950s. His own contributions, as well as further advances made thereafter, are surveyed here with particular emphasis on asymptotic expansions, zeros, inequalities, computational methods, and applications.
Zeros of Dedekind zeta functions in the critical strip
 Math.Comp.66 (1997), 1295–1321. MR 98d:11140 Laboratoire d’Algorithmique Arithmétique, Université BordeauxI,351coursdela Libération, F33405 Talence Cedex France Email address: omar@math.ubordeaux.fr
"... Abstract. In this paper, we describe a computation which established the GRH to height 92 (resp. 40) for cubic number fields (resp. quartic number fields) with small discriminant. We use a method due to E. Friedman for computing values of Dedekind zeta functions, we take care of accumulated roundoff ..."
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Cited by 6 (0 self)
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Abstract. In this paper, we describe a computation which established the GRH to height 92 (resp. 40) for cubic number fields (resp. quartic number fields) with small discriminant. We use a method due to E. Friedman for computing values of Dedekind zeta functions, we take care of accumulated roundoff error to obtain results which are mathematically rigorous, and we generalize Turing’s criterion to prove that there is no zero off the critical line. We finally give results concerning the GRH for cubic and quartic fields, tables of low zeros for number fields of degree 5 and 6, and statistics about the smallest zero of a number field. 0. Introduction and notations The Riemann zeta function and its generalization to number fields, the Dedekind zeta function, have been for well over a hundred years one of the central tools in number theory. It is recognized that the deepest single open problem in mathematics is the settling of the Riemann Hypothesis, and number theorists know that its
The holomorphic flow of the Riemann zeta function
"... The flow of the Riemann zeta function, ˙s = ζ(s), is considered and phase portraits presented. Attention is given to the characterization of the flow lines in the neighborhood of the first 500 zeros on the critical line. All of these zeros are foci. The majority are sources, but in a small proportio ..."
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Cited by 2 (2 self)
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The flow of the Riemann zeta function, ˙s = ζ(s), is considered and phase portraits presented. Attention is given to the characterization of the flow lines in the neighborhood of the first 500 zeros on the critical line. All of these zeros are foci. The majority are sources, but in a small proportion of exceptional cases, the the zero is a sink. To produce these portraits, the zeta function was evaluated numerically to 12 decimal places, in the region of interest, using the Chebyshev method and using Mathematica. The phase diagrams suggest new analytic properties of zeta, a number of which are proved and a number of which are given in the form of conjectures.
Computational Number Theory at CWI in 19701994
, 1994
"... this paper we present a concise survey of the research in Computational ..."
ON SOME RECENT RESULTS IN THE THEORY OF THE ZETAFUNCTION
, 2003
"... This review article is devoted to the Riemann zetafunction ζ(s), defined for ℜ s> 1 as (1.1) ζ(s) = ..."
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This review article is devoted to the Riemann zetafunction ζ(s), defined for ℜ s> 1 as (1.1) ζ(s) =