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

Abstract

The author gives nontrivial upper and lower bounds for the number of primes in the interval $[x-x^{\theta },x]$ for some $0.52⩽\theta ⩽0.525$, 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.

44 pages