Primes in almost all short intervals

TL;DR

This paper improves a result on the distribution of prime numbers, demonstrating that almost all short intervals of a specific size contain primes using advanced sieve techniques and integral estimates.

Contribution

It refines Jia's 1996 result by sharpening the interval size and employing sophisticated analytic methods to show primes appear in almost all such short intervals.

Findings

Almost all intervals of the form [n, n + n^{1/21.5 + ε}] contain primes.

Utilizes Watt's mean value bound and refined sieve decomposition techniques.

Achieves more precise estimates for integrals related to prime distribution.

Abstract

The author sharpens a result of Jia (1996), showing that the interval $[n, n+n^{\frac{1}{21.5}+\varepsilon}]$ contains prime numbers for almost all $n$. Watt's mean value bound, a delicate sieve decomposition and more accurate estimates for integrals are used to good effect.