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A Frobenius determinant proof of the non-existence of Landau–Siegel zeros

A mix of human-written and AI-generated textHuman understanding: no partsmath.NT — Number Theory

Contributed by Roelof van Dijk

Version 2 / Oct 20, 2026 / CC BY 4.0

Abstract

We give a variant of the OpenAI proof [1] of the non-existence of Landau–Siegel zeros, following the simplifications of Kerr [2], Green [3] and Barreto [4]. The Frobenius congruence proves both divisibility and nonvanishing of the determinant. Reduction modulo the largest inert prime below the cutoff reduces nonvanishing to a polynomial root count, replacing resultants and Wronskians. The argument was discovered in discussions with GPT-6 Astra and Opus 5.5 and the new arguments should be attributed to these models.

Tools used

Anthropic
Claude OpusVersion 5.5
OpenAI
ChatGPTVersion 6 Astra

References

  1. OpenAI, Uniform exclusion of Landau--Siegel zeros , OpenAI Math Release, 1 October 2026. https://github.com/openai/math/blob/main/preprints/Uniform-exclusion-of-Landau-Siegel-zeros-October-1-2026/paper.pdf.link
  2. B. Kerr, A simpler proof that there are no Landau--Siegel zeros , arXiv:2610.21099v1, 2026. https://arxiv.org/abs/2610.21099v1.arXiv
  3. B. Green, On Landau--Siegel zeros , hexagon:2610.00311v2, 2026. https://hexagonmath.org/pdf/2610.00311v2.hexagon
  4. K. Barreto, An elementary proof of the non-existence of Landau--Siegel zeros , manuscript. https://x.com/AcerFur/status/2112239827077026128.link
  5. H. L. Montgomery and R. C. Vaughan, Multiplicative Number Theory I: Classical Theory , Cambridge University Press, 2006.
  6. T. Tao, An experimental AI-math challenge: using tenth discriminant bounds to explore arguments ruling out Siegel zeroes , What's New, 20 October 2026. https://terrytao.wordpress.com/2026/10/20/an-experimental-ai-math-challenge-using-tenth-discriminant-bounds-to-explore-arguments-ruling-out-siegel-zeroes/.link

Version history

  1. v1Submitted by Roelof van DijkInitial depositSupersededOct 20, 2026
  2. v2Submitted by Roelof van DijkSubmitted as a new versionCurrentOct 21, 2026