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When dealing with the prime number theorem in arithmetic progressions, one cannot exclude the possible presence of a real zero close to $1$ for at most one real character mod $q$. On the other hand, it is also known that the Riemann $\zeta$ function does not vanish on (0, 1).

Are there any result showing that some Dirichlet L-functions (attached to a non-principal character) do not vanish far from $1$ ? I think non-vanishing at $1/2$ is still open in general, but maybe it is known in some cases.

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The non-vanishing of $L$-series on the real line received a lot of attention, unfortunately, there is still a lot we do not know, even in the non-quadratic case. This circle of problems even has its own MSC number (11M20). In my opinion a good point to start would be the series of article "Elementary methods in the theory of $L$-series" by Pintz.

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  • $\begingroup$ Thank you for the answer, but it seems to me this series of paper is focusing on giving elementary proofs of non-vanishing theorems close to $1$. $\endgroup$ – A. Bailleul Mar 23 at 13:32
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Perhaps the most comprehensive result in this direction is due to Conrey and Soundararajan (https://arxiv.org/abs/math/0111013 or https://link.springer.com/article/10.1007/s00222-002-0227-x). They prove (among other things) that the density of odd positive squarefree integers $d\leq x$ such that $L(s,\chi_{8d})>0$ for all $s\in[0,1]$ is at least $\frac{4}{5\pi^2}$. For work on nonvanishing at $s=1/2$, Soundararajan (https://arxiv.org/abs/math/9902163 or https://www.jstor.org/stable/2661390?seq=1#page_scan_tab_contents) obtains more robust results.

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  • $\begingroup$ Thanks. Both these articles are about density results. I assume there is no known family of Dirichlet L-functions such that no member of this family vanishes on the real line, or even at $1/2$ ? $\endgroup$ – A. Bailleul Mar 23 at 13:35
  • $\begingroup$ @A.Bailleul I do not know of any such result. That seems insurmountable with the current methods. I should mention that density results also exist for the entire family of Dirichlet characters (not just quadratic characters, as considered by the aforementioned results). $\endgroup$ – 2734364041 Mar 24 at 8:05

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