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Mathematics of Computation

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An explicit density estimate for Dirichlet $ L$-series

Author: O. Ramaré
Journal: Math. Comp. 85 (2016), 325-356
MSC (2010): Primary 11P05, 11Y50; Secondary 11B13
Published electronically: June 3, 2015
MathSciNet review: 3404452
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Abstract: We prove that, for $ T\ge 2\,000$, $ T\ge Q\ge 10$, and $ \sigma \ge 0.52$, we have

$\displaystyle \sum _{q\le Q}\mkern -2mu \frac {q}{\varphi (q)}\mkern -6mu \sum ... ...,Q^{5}T^3\bigr )^{1-\sigma }\log ^{5-2\sigma }(Q^2T) \!+\!32\,Q^2\log ^2(Q^2T),$    

where $ \chi \operatorname {mod}^* q$ denotes a sum over all primitive Dirichlet characters $ \chi $ to the modulus $ q$. Furthermore, we have

$\displaystyle N(\sigma ,T,\bbone )\le 2T \log \biggl (1+\frac {9.8}{2T}(3T)^{8(1-\sigma )/{3}}\log ^{5-2\sigma }(T)\biggr ) \!+\!103(\log T)^2.$    

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Additional Information

O. Ramaré
Affiliation: Laboratoire Paul Painlevé / CNRS, Université Lille 1, 59655 Villeneuve d’Ascq, France

Received by editor(s): September 5, 2013
Received by editor(s) in revised form: June 13, 2013, and July 5, 2014
Published electronically: June 3, 2015
Article copyright: © Copyright 2015 American Mathematical Society