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Small algebraic integers and Frobenius prime sums

A mix of human-written and AI-generated textHuman understanding: all partsmath.NT — Number Theory

Contributed by Scott Kominers ↗

Version 1 / Oct 22, 2026 / CC BY-SA 4.0

Abstract

The determinant argument in recent proofs of Landau–Siegel zero exclusion compares the size of a nonzero integer with its divisibility by inert primes. We observe that the same comparison applies to every Frobenius class in a finite Galois extension of Q\mathbb{Q}, bounding its weighted prime count in terms of the supply of small algebraic integers. For each fixed field, the resulting height bound implies a weighted Chebotarev theorem whose only analytic inputs are the elementary Chebyshev and Mertens estimates. In the quadratic case there are only two Frobenius classes, so the lower bound obtained by complementation remains uniform in the conductor; this recovers Landau–Siegel zero exclusion.

Provenance statement

This note was developed in the process of reading and exploring the Green and Barreto arguments with the assistance of GPT-6 Astra (accessed in part via Poe with the support of Quora, where I am an advisor). Claude Opus 5.5 helped identify earlier precedents.

Tools used

OpenAI
ChatGPTVersion 6 Astra
Anthropic
Claude OpusVersion 5.5
Quora
Poe

References

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Version history

  1. v1Submitted by Scott KominersInitial depositCurrentOct 21, 2026