The Cram\'er conjecture holds with a positive probability

TL;DR

This paper demonstrates that there is a positive probability for certain intervals and residue classes to contain prime numbers, supporting conjectures about prime distribution.

Contribution

It proves that a positive proportion of specific intervals and residue classes contain primes, advancing understanding of prime distribution in short intervals and arithmetic progressions.

Findings

A positive proportion of intervals of length proportional to log(X) contain primes.

A positive proportion of residue classes modulo q contain primes less than a multiple of φ(q)log(q).

Supports the Cramér conjecture with probabilistic evidence.

Abstract

We prove that a positive proportion of the intervals of any fixed scalar multiple of $\log(X)$ in the dyadic interval $[X,2X]$ contain a prime number. We also show that a positive proportion of the congruence classes modulo $q$ contain a prime number smaller than any fixed scalar multiple of $\varphi(q)\log(q).$