The number of primes in short intervals and numerical calculations for Harman's sieve

TL;DR

This paper refines bounds on the number of primes in short intervals of the form [x - x^θ, x] for θ around 0.52, confirming the presence of primes in such intervals for large x using advanced sieve techniques.

Contribution

It improves previous bounds on primes in short intervals and confirms their existence for θ approximately 0.52, advancing Harman's sieve methods.

Findings

Interval [x - x^{0.52}, x] contains primes for large x.

Refined bounds for prime counts in short intervals.

Enhanced sieve decomposition techniques.

Abstract

The author gives nontrivial upper and lower bounds for the number of primes in the interval $[x - x^{\theta}, x]$ for some $0.52 \leqslant \theta \leqslant 0.525$, showing that the interval $[x - x^{0.52}, x]$ contains prime numbers for all sufficiently large $x$. This refines a result of Baker, Harman and Pintz (2001) and gives an affirmative answer to Harman and Pintz's argument. New arithmetic information, a delicate sieve decomposition, various techniques in Harman's sieve and accurate estimates for integrals are used to good effect.