An explicit density estimate for Dirichlet $L$-series

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*}