On the differences between consecutive prime numbers, I

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.

6 pages

References in corpus (1)

• The jumping champion conjecture

Cited by in corpus (1)

• On the first occurrences of gaps between primes in a residue class