Negative moments of the gaps between consecutive primes

2018/05/13 by Marek Wolf, Wolf, Marek
Mathematics · #11A41 #11L20 #11N05 #Advanced Mathematical Identities #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT) #math.NT #msc:11A41 #msc:11L20 #msc:11N05

paper · pdf · doi:10.48550/arxiv.1805.04940

Many improvements and changes. Apparently for the first time the analytical formula for the sum of reciprocals of gaps between consecutive primes is presented

openalex publication_date 2018/05/13 · arxiv created 2018/06/06 · arxiv updated 2018/06/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We derive heuristically approximate formulas for the negative k--moments M_-k(x) of the gaps between consecutive primes<x represented directly by π(x) --- the number of primes up to x. In particular we propose an analytical formula for the sum of reciprocals of gaps between consecutive primes <x : ~ M_-1(x)∼ (π²(x))/(x-2π(x))log((x)/(2π(x))) ∼ x log log(x)/log²(x). We illustrate obtained results by the enormous computer data up to x=4× 10¹⁸.

Citations

• Empirical verification of the even Goldbach conjecture and computation of prime gaps up to 4⋅10¹⁸

Related

• On the moments of the gaps between consecutive primes
• Small Gaps Between Primes I
• Positive Proportion of Small Gaps Between Consecutive Primes
• Some heuristics on the gaps between consecutive primes
• Primes in Tuples IV: Density of small gaps between consecutive primes