AI Math

Landau–Siegel zeros and a Stepanov-type nonvanishing lemma

GPT-6 Pro and Claude Opus 5.5 · posted and maintained by Jacob Pescador · v2, 21 October 2026 · unrefereed · PDF · TeX source

Let $\chi$ be a primitive real character of conductor $q\ge 3$. We prove Barreto’s bound

uniformly in $X\ge 3$ and $q$, with an effective implied constant, and hence that every zero $\beta \in (0,1)$ of $L(s,\chi )$ satisfies $(1-\beta )\log q\ge 1-O(\log \log q/\log q)$, so there is no Landau–Siegel zero. Our proof uses a single congruence for both the divisibility and the nonvanishing of an interpolation determinant. As in OpenAI’s proof that such zeros do not exist, the norm of the determinant is bounded above by the size of its entries and below by its divisibility by high powers of the odd inert primes $p$. Here the points are conjugate pairs $(\alpha ,\alpha \mathrm{\prime })$ of quadratic integers, and conjugation agrees with the $p$th power map modulo $p$. Modulo a large inert prime $p_0$, every point lies on the graph $w=z^{p_0}$ of Frobenius, and as $z^{p_0}$ has derivative zero modulo $p_0$, a linear relation among the rows would give a polynomial in $z$ with more roots, counted with multiplicity, than its degree. This is the multiplicity count of Stepanov’s method. The points form Barreto’s rectangular box, which gives the $\frac{1}{4}$.

Versions

• v3 · 22 October 2026 · PDF · TeX
  Says what version 1 introduced, corrects the explanation of why only inert primes count, adds a remark on why the log log X is forced, and adds van Dijk’s class-number-one sieve to the comparison.
• v2 · 21 October 2026 · PDF · TeX
  Following Barreto, a rectangular box of points lowers the coefficient of log q from ½ to ¼ and gives (1−β) log q ≥ 1 − O(log log q / log q). The nonvanishing argument is unchanged, the exposition is rewritten, and the comparison with other proofs adds Barreto’s and van Dijk’s. The introduction now requires p to be odd where v1 did not.
• v1 · 18 October 2026 · PDF · TeX

Cite

GPT-6 Pro and Claude Opus 5.5 (posted by J. Pescador), “Landau–Siegel zeros and a Stepanov-type nonvanishing lemma,” preprint, version 2, 21 October 2026, https://jacobpescador.github.io/ai-math/papers/landau-siegel-stepanov/v2/.

@misc{aimath-landau-siegel-stepanov,
  author       = {{GPT-6 Pro} and {Claude Opus 5.5}},
  title        = {{Landau--Siegel zeros and a Stepanov-type nonvanishing lemma}},
  howpublished = {Preprint, posted by J.~Pescador, \url{https://jacobpescador.github.io/ai-math/papers/landau-siegel-stepanov/v2/}},
  note         = {Version 2, 21 October 2026},
  year         = {2026}
}

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