Real non trivial zeros of Dirichlet L-functions

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.

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.

• 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 $1$. – A. Bailleul Mar 23 at 13:32

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.

• 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 $1/2$ ? – A. Bailleul Mar 23 at 13:35
• @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). – 2734364041 Mar 24 at 8:05