The Prime Number Theorem and Pair Correlation of Zeros of the Riemann Zeta-Function

TL;DR

This paper improves the error bounds in the prime number theorem by leveraging advanced conjectures about the distribution of zeros of the Riemann zeta-function, extending previous results under the Riemann Hypothesis.

Contribution

It introduces a method to refine prime number theorem error estimates using uniform versions of Montgomery's pair correlation conjecture with explicit error terms.

Findings

Error bounds in prime number theorem are improved beyond Riemann Hypothesis limits.

Uniform pair correlation conjectures enable more precise prime distribution estimates.

The approach links zero distribution conjectures to prime number theorem accuracy.

Abstract

We prove that the error in the prime number theorem can be quantitatively improved beyond the Riemann Hypothesis bound by using versions of Montgomery's conjecture for the pair correlation of zeros of the Riemann zeta-function which are uniform in long ranges and with suitable error terms.