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Is there a well known result that states that as $t \to \infty$, 'almost all' zeros of any Dirichlet L function $L(s,\chi)$ lie in the region $R= \{\sigma+i t\mid |\sigma -\frac{1}{2}| \leq \Phi(t) \}$ for a positive function $\Phi$ "slowly going to zero" as $t\to \infty$, in the sense that the limit of the fraction of the # of zeros of the $L$ function, as $t \to \infty$, within $R\cap \{[0,1] \times[-t,t]\}$, is $1$?

In this case, precise quantifiers above would also be needed. I would be grateful for any possible reference (possibly there is a place in Iwaniec-Kowalski's book where this is mentioned precisely).

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    $\begingroup$ Isn't this a form of the Lindelöf hypothesis? I think it is open. $\endgroup$
    – Will Sawin
    Aug 9, 2022 at 16:33
  • $\begingroup$ See exercise 8 in my article on RH, at mltblog.com/3zCsJSz. That's the closest I came to figuring out why there are no zero if $\sigma\geq \frac{1}{2}$, except at $\sigma=\frac{1}{2}$ or $\sigma=1$. It's by no means a proof, but I believe quite insightful. $\endgroup$ Aug 9, 2022 at 17:28
  • $\begingroup$ This is true for $\zeta(s)$, so I suspect it is possible to deduce similar results for fixed $\chi$ or for all $\chi$ associated with a fixed modulus, and I guess proving similar results would be more difficult when $\chi$ is a real character. $\endgroup$
    – TravorLZH
    Aug 9, 2022 at 17:44
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    $\begingroup$ Bohr-Landau theorem? $\endgroup$ Aug 9, 2022 at 18:56

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One way to address this question is via a zero density result, of which there are a great variety. One can find a nice survey of this technology in Chapter 10 of Iwaniec and Kowalski. For instance, their Theorem 10.4 (which they call the Grand Density Theorem) gives a pretty general result that certainly works for an individual Dirichlet $L$-function with some control on the conductor. They also remark that work of Montgomery (Topics in Multiplicative Number Theory) can give some improvements which appear to be desirable based on the statement of the question.

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