# “Sub-logarithmic” zero-free regions from Deuring-Heilbronn/Linnik's repulsion theorem

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", 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, 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", 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 $$q_n$$)?

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.
• Is the $q$ from the exceptional character the same $q$ for the other characters? (This was the exact aspect from Jutila's paper which I did not understand, which was what prompted me to look at Bombieri's "Le Grande Crible") – Alufat Feb 13 at 1:41
• @Alufat: Jutila's Theorem 2 concerns Dirichlet characters of the same modulus $q$. Similarly, Heath-Brown's Theorem 1 concerns a fixed $q$. Of course this theorem is interesting only when infinitely many $q$'s satisfy the conditions, but $q$ is fixed throughout the proof (and $L$ abbreviates its logarithm). In fact on p.206 of Heath-Brown's paper, the $\rho$'s are zeros of the same $L$-function $L(s,\chi)$, where $\chi$ is as in Theorem 1. They serve to approximate $L'(s,\chi)/L(s,\chi)$, cf. (4.1) and (4.2). – GH from MO Feb 13 at 5:46
• @Alufat: I believe that your "Sub-logarithmic ZFR" is harder than Jutila's theorem, and I don't know if it has been deduced from $\delta_n\to 0$. Note, however, that $\delta_n\to 0$ implies $\log q_{n+1}/\log q_n\to\infty$, by Corollary 11.9 in Montgomery-Vaughan: Multiplicative number theory I. – GH from MO Feb 13 at 14:36