On the error in the prime number theorem in short intervals
Number Theory
2025-10-30 v3
Abstract
Assuming the Riemann Hypothesis, we derive explicit bounds for the error terms in short interval analogues of the prime number theorem and Mertens' theorems using a smoothing argument. Our results improve upon previous bounds in both constant terms and applicable ranges, notably improving and extending earlier work by Cully-Hugill and Dudek.
Keywords
Cite
@article{arxiv.2403.11814,
title = {On the error in the prime number theorem in short intervals},
author = {Ethan Simpson Lee},
journal= {arXiv preprint arXiv:2403.11814},
year = {2025}
}
Comments
Major updates to the presentation of the paper and the smoothing argument. The main results of this paper are improved, and the arguments are cleaner. As usual: 15 pages, comments and suggestions are warmly welcomed!