Where can one find a Vinogradov-Korobov zero-free region for Dirichlet L-functions? It has to be in a standard reference, but I'm having a non-trivial time finding it.



Vinogradov-Korobov for Dirichlet L-functions?
My paper <http://arxiv.org/abs/2210.06457> establishes several explicit Vinogradov--Korobov type zero-free regions for Dirichlet $L$-functions. In particular, Theorem 1.1 states the following:


Let $q \geq 3$, and let $\chi\pmod{q}$ be a Dirichlet character. The Dirichlet $L$-function $L(\sigma+it,\chi)$ does not vanish in the region
\begin{equation}\label{smallt}
 \sigma \geq 1-\frac{1}{10.5 \log q+61.5(\log |t|)^{2 / 3}(\log \log |t|)^{1 / 3}}, \quad|t| \geq 10.
\end{equation}
Also, there exists an absolute and effectively computable constant $Y > 0$ such that $L(\sigma+it,\chi)$ does not vanish in the region
\begin{equation}\label{larget}
\sigma \geq 1- \frac{1}{ 10.1\log q + 49.13(\log |t|)^{2/3}(\log\log|t|)^{1/3}},\qquad |t| \geq Y.
\end{equation}


Additionally, a short, self-contained proof of the Vinogradov--Korobov zero-free region for Dirichlet $L$-functions (assuming an explicit upper bound for the Hurwitz zeta function $\zeta(s,u)$ proven by Ford, and a convenient form of Jensen's formula from complex analysis) is provided in the appendix of the paper (albeit with worse constants than Theorem 1.1 above).


