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

DOI10.1007/s11075-014-9893-1zbMath1320.65039arXiv1309.2877OpenAlexW1963479878MaRDI QIDQ2351480

Fredrik Johansson

Publication date: 23 June 2015

Published in: Numerical Algorithms (Search for Journal in Brave)

Full work available at URL: https://arxiv.org/abs/1309.2877

zbMATH Keywords

Hurwitz zeta functionRiemann zeta functionStieltjes constantserror boundpower seriesfast polynomial arithmeticKeiper-Li coefficientsarbitrary-precision arithmeticrigorous numerical evaluation

Mathematics Subject Classification ID

Symbolic computation and algebraic computation (68W30) (zeta (s)) and (L(s, chi)) (11M06) Computation of special functions and constants, construction of tables (65D20) Software, source code, etc. for problems pertaining to number theory (11-04) Euler-Maclaurin formula in numerical analysis (65B15) Hurwitz and Lerch zeta functions (11M35) Numerical approximation and evaluation of special functions (33F05)

Related Items (15)

Acceleration methods for series: a probabilistic perspective ⋮ Error bounds for the asymptotic expansion of the Hurwitz zeta function ⋮ The practical Gauss type rules for Hadamard finite-part integrals using Puiseux expansions ⋮ Highest cusped waves for the Burgers-Hilbert equation ⋮ Series acceleration via negative binomial probabilities ⋮ Fast computation of the Stieltjes constants ⋮ Computing Stieltjes constants using complex integration ⋮ Crandall's computation of the incomplete gamma function and the Hurwitz zeta function, with applications to Dirichlet \(L\)-series ⋮ Approximating and bounding fractional Stieltjes constants ⋮ Computing Hypergeometric Functions Rigorously ⋮ On the Poincaré expansion of the Hurwitz zeta function ⋮ The modified composite Gauss type rules for singular integrals using Puiseux expansions ⋮ Derivatives of \(L\)-series of weakly holomorphic cusp forms ⋮ Discretized Keiper/Li Approach to the Riemann Hypothesis ⋮ Hausdorff dimension estimates applied to Lagrange and Markov spectra, Zaremba theory, and limit sets of Fuchsian groups

Uses Software

• FLINT
• gmp
• Mathematica
• SageMath
• MPFR
• MPIR
• Arb

Cites Work

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