On the number of exceptional intervals to the prime number theorem in short intervals

Abstract

For a fixed exponent $0<\theta \le 1$, it is expected that we have the prime number theorem in short intervals ${\sum }_{x\le n<x+x^{\theta }}\Lambda (n)\sim x^{\theta }$ as $x\to \infty$. From the recent zero density estimates of Guth and Maynard, this result is known for all $x$ for $\theta >\frac{17}{30}$ and for almost all $x$ for $\theta >\frac{2}{15}$. Prior to this work, Bazzanella and Perelli obtained some upper bounds on the size of the exceptional set where the prime number theorem in short intervals fails. We give an explicit relation between zero density estimates and exceptional set bounds, allowing for the most recent zero density estimates to be directly applied to give upper bounds on the exceptional set via a small amount of computer assistance.

17 pages