Numerical Computations Concerning the GRH

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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 languageEnglish
Pages (from-to)3009-3027
Number of pages19
JournalMathematics of Computation
Issue number302
Early online date15 Jan 2016
Publication statusPublished - Nov 2016


  • Primary 11M26
  • Primary 11M06
  • Secondary 11P32


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  • HPC (High Performance Computing) Facility

    Susan L Pywell (Manager), Simon A Burbidge (Other), Polly E Eccleston (Other) & Simon H Atack (Other)

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