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

14 Citations (Scopus)

328 Downloads (Pure)

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

Access to Document

10.1090/mcom/3077

Full-text PDF (accepted author manuscript)
First published in Mathematics of Computation in 2015, published by the American Mathematical Society.
Accepted author manuscript, 332 KB

HPC (High Performance Computing) Facility
Susan L Pywell (Manager), Simon A Burbidge (Other), Polly E Eccleston (Other) & Simon H Atack (Other)
Facility/equipment: Facility

Cite this

@article{e58df5f156e24e6daa1942584e5c02bb,

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.",

keywords = "Primary 11M26, Primary 11M06, Secondary 11P32",

author = "Platt, {David J}",

year = "2016",

month = nov,

doi = "10.1090/mcom/3077",

language = "English",

volume = "85",

pages = "3009--3027",

journal = "Mathematics of Computation",

issn = "0025-5718",

publisher = "American Mathematical Society",

number = "302",

}

TY - JOUR

T1 - Numerical Computations Concerning the GRH

AU - Platt, David J

PY - 2016/11

Y1 - 2016/11

N2 - 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.

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)

VL - 85

SP - 3009

EP - 3027

JO - Mathematics of Computation

JF - Mathematics of Computation

SN - 0025-5718

IS - 302

ER -

Numerical Computations Concerning the GRH

Abstract

Keywords

Access to Document

Fingerprint

Equipment

HPC (High Performance Computing) Facility

Cite this