On the OpenAI proof that Landau-Siegel zeros do not exist

Abstract

A recent manuscript of OpenAI asserts that Landau-Siegel zeros do not exist. Here, we give a detailed account of its argument, arranged as a direct proof with explicit constants. There are absolute constants $q_0$ and ${\delta }_0>0$ such that, for every primitive real Dirichlet character $\chi$ of conductor $q\ge q_0$, every real zero $\beta \in (0,1)$ of $L(s,\chi )$ satisfies $(1-\beta )\log q\ge {\delta }_0$, and one can take $q_0=10^{10}$ and ${\delta }_0=3.0\times 10^{-3}$. Combined with a recent computation of Lu, Zaman, and Zhao, this result yields the bound $(1-\beta )\log q\ge 3.0\times 10^{-3}$ for every conductor $q\ge 3$. The proof is an interpolation determinant argument in the biquadratic field generated by $\sqrt{2}$ over the quadratic field attached to $\chi$. A real zero close to one forces $\chi (p)=-1$ for most primes $p$, and the Frobenius congruence at these primes makes a certain nonzero determinant divisible by higher powers of these primes than its Archimedean size allows. The underlying ideas are those of the OpenAI manuscript. The explicit constants, and the presentation, are new.

arXiv comment19 pages; 1 figure; comments welcome!