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arXiv:2211.02515math.NT

Discrete mean estimates and the Landau-Siegel zero

Yitang Zhang

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Abstract

Let $χ$ be a real primitive character to the modulus $D$. It is proved that $$ L(1,χ)\gg (\log D)^{-2022} $$ where the implied constant is absolute and effectively computable. In the proof, the lower bound for $L(1,χ)$ is first related to the distribution of zeros of a family of Dirichlet $L$-functions in a certain region, and some results on the gaps between consecutive zeros are derived. Then, by evaluating certain discrete means of the large sieve type, a contradiction can be obtained if $L(1,χ)$ is too small.

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math.NTNumber Theory

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Linked by 1 identifiable commenter in 1 comment across 1 thread, from Nov 2022 to Nov 2022. Found in 2 HN submissions from Nov 2022 to Nov 2022. Open submission discussions.

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  1. Monumental (if correct) advance in number theory posted to ArXiv by Yitang Zhang 1 comment

    • arXiv https://arxiv.org/abs/2211.02515 Discussion https://old.reddit.com/r/math/comments/y93a86/eliundergradua... https://mathoverflow.net/questions/433949/consequences-resul... New Yorker article https://www.newyorker.com/magazine/2015/02/02/pursuit-beauty

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  • Monumental (if correct) advance in number theory posted to ArXiv by Yitang Zhang1303 points417 comments

    Yitang Zhang, the mathematician behind the 2013 breakthrough on bounded gaps in primes, posted to the arxiv today a result which (if correct) comes close to proving the nonexistence of Landau--Siegel zeros: https://arxiv.org/abs/2211.02515. To give a sense of the scale of this claim: If correct, Zhang's work is the most significant progress towards the Generalized Riemann Hypothesis in a century. Moreover, I think this result would not only be a more significant advance than Zhang's previous…

  • Discrete mean estimates and the Landau-Siegel zero3 points0 comments

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