# What are the best known upper bounds for $\frac{1}{L(s, \chi)}$?

Let $$\chi$$ be a Dirichlet character and $$L(s, \chi)$$ be the corresponding Dirichlet L-function. What are the best known bounds for $$\frac{1}{L(s, \chi)}$$ in the half-plane of convergence?

I'm aware of the bounds near $$\Re(s)=1$$, from which one can deduce the non vanishing of $$L(s, \chi)$$ there. However, I'm yet to see anything in the literature pertaining to my question.

• If you don't have a zero-free region, you shouldn't be able to bound 1/L(s, \chi). In what region do you want bounds? Mar 27 at 22:50
• @MayankPandey, like I said, I'm looking for bounds in the half-plane of convergence. That is, the region $\Re(s) >c_{\chi}$ where $c_{\chi} \in \mathbb{R}$ is the minimal constant such that $L(s, \chi) \neq 0$ for $\Re(s)>c_{\chi}$. Mar 27 at 22:55

For unconditional results, see Theorem 11.4 of Montgomery-Vaughan, which states the following. Let $$\chi$$ be a primitive Dirichlet character modulo $$q > 1$$. Then there exists a constant $$c > 0$$ such that $$L(s,\chi)$$ has at most one zero in the region $$\Re(s) > 1 - \frac{c}{\log(q(|\Im(s)| + 3))},$$ and this exceptional zero can only exist if $$\chi$$ is quadratic, in which case this zero is real. (This is the standard zero-free region for Dirichlet $$L$$-functions.) We denote by $$\beta$$ such an exceptional zero (which is called a Landau-Siegel zero). Then if $$s$$ is in this region and additionally if $$|s - \beta| > 1/\log q$$ if $$\beta$$ exists, we have that $$\frac{1}{L(s,\chi)} \ll \log(q(|\Im(s)| + 3)).$$ If $$\beta$$ exists, $$s$$ is in this region, and additionally if $$0 < |s - \beta| \leq 1/\log q$$, then $$\frac{1}{|s - \beta| (\log q)^2} \ll \frac{1}{|L(s,\chi)|} \ll \frac{1}{|s - \beta|}.$$
• If $|s-\beta|>1/\log q$, then the stronger upper bound $\ll \log q+(\log(|\Im(s)|+3))^{2/3}(\log\log(|\Im(s)|+3))^{1/3}$ holds because of the Vinogradov-Korobov zero-free region. Mar 28 at 2:58
• @2734364041 Is this written down anywhere? The zero-free region is in Tanmay Khale's paper, but I don't know a reference for where this is translated into bounds for $L(s,\chi)$. Mar 28 at 17:37
• Peter: That might not be written anywhere. But one start with Proposition 5.16 in Iwaniec-Kowalski, estimate the sum over zeros using Khale's zero-free region, and integrate to obtain a strong bound on $|\log L(1,\chi)|$. Mar 29 at 2:53
• Is there a link between the fact that the absolute Galois group of $\mathbb{R}$ is of order $2$ and the quadraticity of $\chi$? Mar 31 at 16:10