Vinogradov-Korobov for Dirichlet L-functions?

Question

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.

Answer 1

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).

Answer 2

Languasco and Zaccagnini cite Prachar, and mention Vasil'kovskaja. https://core.ac.uk/reader/81188410 https://doi.org/10.1016/j.jnt.2006.12.015

They give a zero-free region in Lemma 1 (suppressing details).

See also this question from a decade ago. Mertens-like sum in arithmetic progressions

And this one from last year (answered by Languasco): Error term in Mertens' third theorem

Answer 3

In Theorem 2 of Mark Coleman's paper below, a full proof of the Vinogradov-Korobov zero-free region for the $L$-function of a Grossencharacter twisted by a Hecke character over an arbitrary number field is provided. Restricting to $K=\mathbb{Q}$, one recovers the result for Dirichlet $L$-functions. This is the only place where I have seen a full proof of such a zero-free region published.

Coleman, M. D., A zero-free region for the Hecke L-function, Mathematika 37, No. 2, 287-304 (1990). ZBL0721.11050.