# Mathematics of Computation

Published by the American Mathematical Society since 1960 (published as Mathematical Tables and other Aids to Computation 1943-1959), Mathematics of Computation is devoted to research articles of the highest quality in computational mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.78.

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## An explicit density estimate for Dirichlet $L$-seriesHTML articles powered by AMS MathViewer

by O. Ramaré
Math. Comp. 85 (2016), 325-356 Request permission

## Abstract:

We prove that, for $T\ge 2\,000$, $T\ge Q\ge 10$, and $\sigma \ge 0.52$, we have \begin{equation*} \sum _{q\le Q}\mkern -2mu \frac {q}{\varphi (q)}\mkern -6mu \sum _{\chi \operatorname {mod}^* q}\mkern -12mu N(\sigma ,T,\chi ) \!\le \! 20\bigl (56\,Q^{5}T^3\bigr )^{1-\sigma }\log ^{5-2\sigma }(Q^2T) \!+\!32\,Q^2\log ^2(Q^2T), \end{equation*} where $\chi \operatorname {mod}^* q$ denotes a sum over all primitive Dirichlet characters $\chi$ to the modulus $q$. Furthermore, we have \begin{equation*} N(\sigma ,T,\mathbb {1} )\le 2T \log \biggl (1+\frac {9.8}{2T}(3T)^{8(1-\sigma )/{3}}\log ^{5-2\sigma }(T)\biggr ) \!+\!103(\log T)^2. \end{equation*}
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