Numerical computations concerning the GRH

Platt, David

doi:10.1090/mcom/3077

Numerical computations concerning the GRH
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by David J. Platt PDF

Math. Comp. 85 (2016), 3009-3027 Request permission

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 $q\leq 400 000$. We check, to height, $\textrm {max}\left (\frac {10^8}{q},\frac {A\cdot 10^7}{q}+200\right )$ 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 $q\leq 2 000 000$ vanishes at its central point.

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