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Goldbach numbers in intervals of length 4.45log⁡2x4.45 \log^2 x under the Riemann Hypothesis

Primarily AI-generated textHuman understanding: some partsmath.NT — Number Theory

Contributed by Steve Hubbard ↗

Version 1 / Oct 26, 2026 / CC BY 4.0

Abstract

Assuming the Riemann Hypothesis, we prove that for every real x≥3x \ge 3 the interval (x,x+4.45log⁡2x)(x, x + 4.45 \log^2 x) contains an even integer that is a sum of two odd primes. The proof combines a weighted mean square bound for the Chebyshev prime counting error in short intervals with a piecewise linear weight in a bilinear identity for short interval prime counts. The deduction from explicit analytic inputs, stated as hypotheses, is checked in Lean 4.

Provenance statement

The result came out of an AI-run Lean 4 lab that formalizes Goldbach-related papers. Starting from the explicit RH argument of Chirre and Hagen, arXiv 2512.23534, AI agents found a sharper short interval mean square constant, 1.0069 against 2.2258, and a changed piecewise linear weight in the bilinear pairing, giving 4.45 in place of 123. The deduction is checked in Lean 4 with Mathlib: the main theorem goldbach_445_hsw_gZ builds with no sorry and only the standard axioms, conditional on named cited inputs stated as hypotheses. Gemini drafted the text over ten feedback rounds, the last edits by Claude; Claude checked each round against the Lean statements and the cited sources. The author read the paper and directed the work but has not refereed the analysis. Also posted at aiXiv:2610.02672.

Tools used

Google
GeminiVersion Argon High
Anthropic
Claude Opus
Lean
LeanVersion Lean 4 with Mathlib

References

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Show all 11 referencesShow fewer references
  1. E. C. Titchmarsh, The Theory of the Riemann Zeta-Function , 2nd ed., revised by D. R. Heath-Brown, Oxford University Press, New York, 1986.

Version history

  1. v1Submitted by Steve HubbardInitial depositCurrentOct 25, 2026