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Numerical computations concerning the GRH

Author: David J. Platt
Journal: Math. Comp. 85 (2016), 3009-3027
MSC (2010): Primary 11M26, 11M06; Secondary 11P32
Published electronically: January 15, 2016
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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\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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Additional Information

David J. Platt
Affiliation: Heilbronn Institute for Mathematical Research, University of Bristol, University Walk, Bristol, BS8 1TW, United Kingdom

Received by editor(s): February 8, 2014
Received by editor(s) in revised form: February 20, 2015, and May 11, 2015
Published electronically: January 15, 2016
Article copyright: © Copyright 2016 American Mathematical Society