For each $n\in\mathbb{N}$, let:


* $\chi\_n\pmod{q\_n}$ a real non-principal Dirichlet character ($q\_1 < q\_2 < \cdots$),
* $\beta\_n$ the largest real zero of $L(s,\chi\_n)$,
* $\delta\_n := (1-\beta\_n)\log(q\_n)$.


Let $\chi\pmod{q}$ be a Dirichlet character, and consider $s = \sigma + it$ with $|t| < 1$. In p. 206 of Heath-Brown's ["Prime twins and Siegel zeros"](http://www.ams.org/mathscinet-getitem?mr=703977), it is mentioned that the *Deuring-Heilbronn phenomenon* implies that there is some absolute constant $C>0$ such that, for each $n\in \mathbb{N}$, the region
$$ \bigg\{ s ~\bigg|~ \sigma \geq 1 - \frac{C\log(\delta\_n^{-1})}{\log(q)},\ |t| < 1 \bigg\} $$
has no zeros (besides $\beta\_1,\ldots,\beta\_n$) of $L(s,\chi)$. (At least, that is how I interpreted the assertion "$r\_0 \gg L^{-1}\log \eta$" at the mentioned page). Assuming this statement, it follows that:



> 
> **(Sub-logarithmic zero-free regions (ZFR))** If there exists a sequence of Siegel zeros $\beta\_n$ with $\delta\_n \to 0$, then all the other zeros $\sigma + i\gamma$ of $L(s,\chi)$ with $|\gamma| < 1$ for Dirichlet characters $\chi\pmod{q}$ satisfy:
>  $$ \sigma < 1 - \frac{1}{o(\log(q))}. $$
> 
> 
> 


It appears to me that this (or slight variations of this) statement is often used in the literature [the only example I have in mind at the moment is Remark 1 in p. 515 (p. 6 in the link) of Granville & Stark's [$ABC$ implies no "Siegel zeros" for $L$-functions of characters of negative discriminant](https://www.dms.umontreal.ca/~andrew/PDF/NoSiegelfinal.pdf), where it is mentioned that $\delta\_n \to 0$ implies $\frac{L'}{L}(1,\chi\_n) = (1-\beta\_n)^{-1} + o(\log(q\_n))$].




---


However, I am having trouble following the deduction of these "sub-logarithmic ZFRs" from the Deuring-Heilbronn phenomenon alone. Using the Deuring-Heilbronn (Linnik's repulsion theorem) as in Théorème 16, Sec. 6 of Bombieri's ["Le grande crible"](http://www.numdam.org/issue/AST_1987__18__1_0.pdf), we get that there are absolute constants $c\_1,c\_2 >0$ such that, fixing $n\in\mathbb{N}$, it holds:
$$ \sigma < 1 - c\_1\log\left(c\_2\frac{\delta\_n^{-1}}{\log(q\_n q)/\log(q\_n)}\right) \cdot \frac{1}{\log(q\_n q)} $$
(I am just taking $T = q\_n q$ in Théorème 16). Assuming we have an infinite sequence $q\_n \to +\infty$ with $\delta\_n \to 0$, for a given $q\in\mathbb{N}$ we may take $q\_k \leq q < q\_{k+1}$, so that $\log(q\_{k+1} q)/\log(q\_{k+1})< 2$, and hence:
$$ \sigma < 1 - c\_1\frac{\log(\frac{c\_2}{2}\delta\_{k+1}^{-1})}{\log(q\_{k+1} q)}. $$


It appears, then, that to derive the "sub-logarithmic" ZFRs, it is necessary to have $\log(q\_{k+1}) \ll \log(q\_k)$ as $k\to \infty$, i.e.: *the gaps between the conductors of consecutive exceptional characters need to be polynomially bounded, even if we assume $\delta\_n \to 0$*.


I do not think my conclusion is correct (e.g., I believe I misinterpreted some aspect of Heath-Brown's paper), but I have not been able to get rid of this condition on the growth of the $q\_n$. In short, my question is the following:



> 
> **Q.** Is it really possible to derive "sub-logarithmic" ZFRs from Linnik's repulsion theorem (i.e., without additional growth conditions on the $q\_n$)?
> 
> 
> 



"Sub-logarithmic" zero-free regions from Deuring-Heilbronn/Linnik's repulsion theorem
Let $\chi$ be a non-principal real Dirichlet character modulo $q$. Let 
$$\beta\_0=1-\frac{1}{\eta\log q}$$
be a real zero of $L(s,\chi)$ satisfying $\eta\geq 100$ for convenience (Heath-Brown's condition is $\eta\geq 3$). Let $\rho=\beta+i\gamma$ be any zero of $L(s,\chi)$ such that $\rho\neq\beta\_0$ and $|\gamma|\leq 1$. We strengthen Heath-Brown's claim $r\_0\gg L^{-1}\log\eta$ to (note that $L=\log q)$
$$1-\beta\gg\frac{\log\eta}{\log q}.$$
We shall deduce this from Theorem 2 of Jutila's 1977 paper "On Linnik's constant", which is also Heath-Brown's reference. Let us write $\delta$ for the left hand side. If $\delta>1/60$, then the statement is trivial by $\eta\ll q$. So we shall assume that $\delta\leq 1/60$. Then, Jutila's theorem yields readily (using $D\leq 2q$) that
$$1-\beta\_0\geq\frac{1}{10}\cdot\frac{q^{-3\delta}}{\log q}.$$
In other words, $\eta\leq 10 q^{3\delta}$. By the assumption $\eta\geq 100$, this implies that $\eta\leq q^{6\delta}$, or equivalently that
$$\delta\geq\frac{1}{6}\cdot\frac{\log\eta}{\log q}.$$
We have verified Heath-Brown's claim.


