Landau–Siegel zeros and a Stepanov-type nonvanishing lemma

GPT-6 Pro and Claude Opus 5.5 · posted and maintained by Jacob Pescador · v1, 18 October 2026 · unrefereed · PDF · TeX source

Let

χ

be a primitive real Dirichlet character of conductor

q

. We show that

\sum_{p \leq X, χ (p) = - 1} \frac{\log p}{p} \leq \frac{1}{2} \log X + \frac{1}{2} \log q + \log \log X + O (1)

\begin{matrix} \sum_{p \leq X, χ (p) = - 1} \frac{\log p}{p} \leq \frac{1}{2} \log X + \frac{1}{2} \log q \\ + \log \log X + O (1) \end{matrix}

uniformly in

X \geq 3

and

q

; consequently

L (s, χ)

has no Landau–Siegel zero. This is a variant of the OpenAI proof, following Green and Kerr. The key arithmetic observation is that Frobenius at a prime inert in the quadratic field of

χ

is conjugation. At the small inert primes it makes a certain algebraic integer highly divisible; at one larger inert prime, through a short Stepanov-type multiplicity argument that replaces the resultant and zero-estimate arguments of Green and Kerr, it shows that this integer is nonzero. This manuscript was written entirely by AI systems; see the note at the end.

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 1, 18 October 2026, https://jacobpescador.github.io/ai-math/papers/landau-siegel-stepanov/v1/.

BibTeX

@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/v1/}},
  note         = {Version 1, 18 October 2026},
  year         = {2026}
}

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