On the differences between consecutive prime numbers, I

2011/11/14 by D. A. Goldston, Goldston, D. A., Andrew Ledoan +1
Mathematics · #11N05 #11N36 #11P32 #Advanced Mathematical Identities #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.1111.3380

openalex publication_date 2011/11/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We show by an inclusion-exclusion argument that the prime k-tuple conjecture of Hardy and Littlewood provides an asymptotic formula for the number of consecutive prime numbers which are a specified distance apart. This refines one aspect of a theorem of Gallagher that the prime k-tuple conjecture implies that the prime numbers are distributed in a Poisson distribution around their average spacing.

Related

• Distribution of Large Gaps Between Primes
• Asymptotic of the greatest distance between adjacent primes and the Hardy-Littlewood conjecture
• Primes in Tuples I
• On the frequency of small gaps between the primes
• A note on Primes in Short Intervals