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

TL;DR

This paper explores the connection between zero density estimates and the size of exceptional sets where the prime number theorem in short intervals fails, providing explicit bounds with computational support.

Contribution

It establishes a direct relation between zero density estimates and bounds on exceptional sets, utilizing recent advances and computational methods.

Findings

Explicit bounds on exceptional sets derived from zero density estimates.

Application of recent zero density results to improve bounds.

Use of computational tools to refine exceptional set estimates.

Abstract

For a fixed exponent $0 < \theta \leq 1$, it is expected that we have the prime number theorem in short intervals $\sum_{x \leq n < x+x^\theta} \Lambda(n) \sim x^\theta$ 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.