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hexagon:2610.00560v1

math.NT — Number Theory

A lower explicit threshold for Chen's theorem under GRH

Contributed by Steve Hubbard

Assume the Generalized Riemann Hypothesis (GRH). We show that every even integer N≥exp⁡(exp⁡(13.375))N \ge \exp(\exp(13.375)) can be written as N=p+P2N = p + P_2, where pp is a prime and P2P_2 is either a prime or a product of two primes. The result is conditional on GRH and on Theorems 5, 6, 7 and Lemma 9 of Bordignon and Starichkova, Ramanujan J. 64 (2024), together with the auxiliary lemmas, Table 1 and display (6) that their Section 5.4 uses, at new parameter values; they obtain exp⁡(exp⁡(14))\exp(\exp(14)). It is not a new method: it is a re-choice of the parameters uu and α\alpha in their argument, combined with the observation that Chen's theorem needs only that the Section 5.4 main-term combination be positive, not that it exceed 4⋅10−34\cdot 10^{-3}, because the inequality π2′(N)≥π2(N)−N−1\pi_2'(N) \ge \pi_2(N) - \sqrt{N} - 1 from the proof of their Corollary 1 gives N=p+P2N = p + P_2 as soon as π2(N)>N+1\pi_2(N) > \sqrt{N} + 1. The arithmetic of Section 5.4 at the new parameters and the parameter range inequalities are verified unconditionally in Lean 4.

Primarily AI-generated textHuman understanding: some partsChen's theoremGeneralized Riemann HypothesisLean 4explicit boundssieve methods
Oct 26, 2026PDF ↗View work →
hexagon:2610.00542v1

math.NT — Number Theory

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

Contributed by Steve Hubbard

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.

Primarily AI-generated textHuman understanding: some partsGoldbach numbersLean 4Riemann Hypothesisexplicit boundsshort intervals
Oct 26, 2026PDF ↗View work →