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I have couple questions regarding the exceptional zeros of Dirichlet $L$-functions. We have the following result: There is a constant $c_1 > 0$ such that $L(\sigma, \chi) \not = 0$ whenever $$ \sigma \geq 1 - \frac{c_1}{\log P}, $$ for all primitive characters $\chi$ modulus $q \leq P$, with the possible exception of at most one primitive character $\tilde{\chi} (mod \ \tilde{r})$. If it exists, the character $\tilde{\chi}$ is quadratic, and the unique exceptional real zero $\tilde{\beta}$ of $L(s, \tilde{\chi})$ satisfies $$ \frac{c_2}{\tilde{r}^{1/2} \log^2 \tilde{r}} \leq 1 - \tilde{\beta} \leq \frac{c_1}{\log P}. $$

I was wondering about the following.

1) Is there a lower bound for $\tilde{r}$?

2) I have seen in couple places without explanation that $(1 - \tilde{\beta}) \log P$ tends to $0$. How does one deduce to this?

Thank you very much.

PS I would like to change my question as the two questions have been answered in the comments. I would greatly appreciate if someone could explain me or provide reference for how to prove the inequality $\frac{c_2}{\tilde{r}^{1/2} \log^2 \tilde{r}} \leq 1 - \tilde{\beta}$ as I am having difficulty finding a reference for this.

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    $\begingroup$ 1) In your second display you have written down a trivial lower bound for $\tilde{r}$ in terms of $P$. This is very weak but explicit, while Siegel's theorem leads to the stronger but ineffective bound $\tilde{r} \gg_A (\log{P})^A$. 2) There are no such results. Rather, the usual definition of Siegel zeros is as a sequence of such $\tilde{\beta}$ having $(1-\tilde{\beta}) \log{P} \to 0$. $\endgroup$ Dec 27, 2017 at 12:59
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    $\begingroup$ Continuing Vesselin Dimitrov's comment, the reason that that is the usual definition is that if there were no such subsequence where $(1-\tilde \beta)\log P\to0$, then one could decrease the value of $c_1$ to the point where there were no exceptional zeros at all. This is the reason that it doesn't make sense to speak of exceptional zeros except in the context of a family of $L$-functions (where the constant $c_1$ is uniform over the whole family; if it's allowed to depend upon a particular modulus, then there will never be an exceptional zero). $\endgroup$ Dec 27, 2017 at 15:59
  • $\begingroup$ Thank you for your comments I have changed the question slightly now. $\endgroup$
    – Johnny T.
    Dec 30, 2017 at 12:31
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    $\begingroup$ It is in all the standard textbooks. For instance, chapter 14 of Davenport's book. The $\log^2$ may actually be removed relatively straightforwardly, cf. Goldfeld and Schinzel's paper archive.numdam.org/article/ASNSP_1975_4_2_4_571_0.pdf . Siegel gives you $1-\tilde{\beta} \gg_A \tilde{r}^{-1/A}$ for every $A < \infty$, implying $\tilde{r} \gg_A (\log{P})^A$ (but ineffectively). $\endgroup$ Dec 30, 2017 at 14:55
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    $\begingroup$ See also mathoverflow.net/questions/211034/… . $\endgroup$ Dec 30, 2017 at 14:57

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