Best known zero-free region for Dirichlet $L$-functions in the $q$-aspect

Question

It is classical that there is a $c > 0$ such that for all Dirichlet characters $\chi$ except for at most one exception, one has that $L(s,\chi)$ has no zeroes for $\sigma > 1 - \frac{c}{\log{q} + \log{(t+2)}}$, where $q$ is the conductor of $\chi$ and $s = \sigma + it$. What is the state of the art?

For instance, is it even known that one has $1 - \sigma > \omega(\log{(q(t+2))}^{-1})$? (That is, that one can change the "there exists $c > 0$" to a "for all $c > 0$". Of course if there is a sufficiently 'bad' Siegel zero then Deuring-Heilbronn says yes, but hopefully something more is known.)

I know there is a Vinogradov-Korobov-type estimate available as $t$ grows and $q$ is fixed, but I couldn't find anything better than the classical bound as $q$ grows. I only really care about the case of real zeroes and quadratic characters, but of course this is the hardest case.

Sorry for the spam of questions about Siegel zeroes, I am very much a nonexpert.

Answer 1

No, such a result is not known unconditionally, even if one restricts to the low-lying case $t=O(1)$ and excludes real zeroes, and it would be a great breakthrough if one could do this. If one had such an improved zero-free region, one could for instance resolve Vinogradov's conjecture on the least quadratic nonresidue (e.g. by applying the results of Granville and Soundararajan, see Theorem 1.6 of http://arxiv.org/abs/1501.01804, in the contrapositive; one can also use the older results of Rodosskii, http://www.ams.org/mathscinet-getitem?mr=82521), which is currently open. Indeed the problem of enlarging the zero-free region in the q-aspect is very closely tied to the problem of estimating short character sums, in much the same way that the Vinogradov-Korobov type enlargements to the zero-free region in the t-aspect are tied to the problem of estimating short exponential sums.

On the other hand, we do have the Deuring-Heilbronn repulsion effect, which shows that the presence of an exceptional zero does allow one to enlarge the classical zero-free region in more or less the manner requested; see e.g. Chapter 18 of Iwaniec-Kowalski. Of course this improvement is "illusory" in the sense that we don't actually believe that exceptional zeroes exist, but this result is still important for obtaining unconditional proofs of results such as Linnik's theorem on the least prime in an arithmetic progression.

ADDED LATER: it may also be possible to get better zero-free regions for those characters for which improved character sum estimates are known, such as when the modulus is very smooth or the character has a small odd order. I don't know if this has been explicitly addressed in the literature though.