Numerical Computations Concerning the GRH

David J Platt

doi:10.1090/mcom/3077

Numerical Computations Concerning the GRH

David J Platt

Research output: Contribution to journal › Article (Academic Journal) › peer-review

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Abstract

We describe two new algorithms for the efficient and rigorous computation of Dirichlet L-functions and their use to verify the Generalised Riemann Hypothesis for all such L-functions associated with primitive characters of modulus q ≤ 400000. We check, to height, max ({10⁸/q},{A.10⁷/q}+200) with A=7.5 in the case of even characters and A=3.75$ for odd characters. In addition we confirm that no Dirichlet L-function with a modulus q ≤ 2000000 vanishes at its central point.

Original language	English
Pages (from-to)	3009-3027
Number of pages	19
Journal	Mathematics of Computation
Volume	85
Issue number	302
Early online date	15 Jan 2016
DOIs	https://doi.org/10.1090/mcom/3077
Publication status	Published - Nov 2016

Keywords

Primary 11M26
Primary 11M06
Secondary 11P32

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10.1090/mcom/3077

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First published in Mathematics of Computation in 2015, published by the American Mathematical Society.
Accepted author manuscript, 332 KB

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Cite this

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title = "Numerical Computations Concerning the GRH",

abstract = "We describe two new algorithms for the efficient and rigorous computation of Dirichlet L-functions and their use to verify the Generalised Riemann Hypothesis for all such L-functions associated with primitive characters of modulus q ≤ 400000. We check, to height, max (\{108/q\},\{A.107/q\}+200) with A=7.5 in the case of even characters and A=3.75\$ for odd characters. In addition we confirm that no Dirichlet L-function with a modulus q ≤ 2000000 vanishes at its central point. ",

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AB - We describe two new algorithms for the efficient and rigorous computation of Dirichlet L-functions and their use to verify the Generalised Riemann Hypothesis for all such L-functions associated with primitive characters of modulus q ≤ 400000. We check, to height, max ({108/q},{A.107/q}+200) with A=7.5 in the case of even characters and A=3.75$ for odd characters. In addition we confirm that no Dirichlet L-function with a modulus q ≤ 2000000 vanishes at its central point.

KW - Primary 11M26

KW - Primary 11M06

KW - Secondary 11P32

U2 - 10.1090/mcom/3077

DO - 10.1090/mcom/3077

M3 - Article (Academic Journal)

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VL - 85

SP - 3009

EP - 3027

JO - Mathematics of Computation

JF - Mathematics of Computation

IS - 302

ER -

Numerical Computations Concerning the GRH

Abstract

Keywords

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