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An explicit elementary proof of no Landau--Siegel zeros

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

Contributed by Victor Z. Guo ↗

Version 1 / Oct 22, 2026 / CC BY 4.0

Abstract

We give an elementary proof of the absence of Landau--Siegel zeros with the explicit constant c=1/100c=1/100. For every primitive quadratic Dirichlet character χ\chi of conductor q≥3q\ge3, each real zero β∈(0,1)\beta\in(0,1) of L(s,χ)L(s,\chi) satisfies (1−β)log⁡q>c(1-\beta)\log q>c. The proof combines integer coefficient counting, polynomial congruences, a paired-tail hyperbola estimate, and a logarithmically weighted comparison of positive sums. All required constants are explicit, and one choice of parameters covers every conductor. The argument uses elementary arithmetic and real summation and integration, without separate zero computations for small conductors.

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References

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Version history

  1. v1Submitted by Victor Z. GuoInitial depositCurrentOct 22, 2026