Abstract
In this article, we investigate the quantity , as over the primes, where is the Dirichlet L-function attached to a non-principal Dirichlet character modulo q. Our main result shows that . We also compute mq for every odd prime q up to 10. As a consequence we numerically verified that for every prime q, , we have , with . In particular, this shows that for every non-principal Dirichlet character χ mod q where is prime, answering a question of Gun, Murty and Rath in this range. We also provide some statistics and scatter plots regarding the mq-values. The programs used and the computational results described here are available at the following web address: http://www.math.unipd.it/∼languasc/smallvalues.html.
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2010 Mathematics Subject Classification:
Acknowledgments
We thank to the referees for their remarks and suggestions. The second author (A. Languasco) would like to thank Luca Righi (University of Padova) for his help in organizing the computation described in Section 5 on the University of Padova Strategic Research Infrastructure Grant 2017: “CAPRI: Calcolo ad Alte Prestazioni per la Ricerca e l’Innovazione”, http://capri.dei.unipd.it.
Declaration of Interest
No potential conflict of interest was reported by the author(s).