An asymptotic upper bound on prime gaps

2015/06/10 by André LeClair, LeClair, André
Mathematics · Physics and Astronomy · #Analytic Number Theory Research #FOS: Mathematics #FOS: Physical sciences #Finite Group Theory Research #Limits and Structures in Graph Theory #Mathematical Physics (math-ph) #Number Theory (math.NT) #math-ph #math.MP #math.NT

paper · pdf · doi:10.48550/arxiv.1506.03359

The proof of the last theorem of the last version is incorrect, because the fluctuations were treated too smoothly. We could only replace it with a weaker result

openalex publication_date 2015/06/10 · arxiv created 2015/10/07 · arxiv updated 2015/10/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The Cramér-Granville conjecture is an upper bound on prime gaps, g_n = p_n+1 - p_n < \cCramer log² p_n for some constant \cCramer ≥ 1. Using a formula of Selberg, we first prove the weaker summed version: ∑_n=1^N g_n < ∑_n=1^N log² p_n. In the remainder of the paper we investigate which properties of the fluctuations \fluc (x) = π(x) - \Li(x) would imply the Cramér-Granville conjecture is true and present two such properties, one of which assumes the Riemann Hypothesis; however we are unable to prove these properties are indeed satisfied. We argue that the conjecture is related to the enormity of the Skewes number.

Citations

• Estimates of Some Functions Over Primes without R.H.

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• Gaps Between Consecutive Primes and the Exponential Distribution
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