Oscillations of the error term in the prime number theorem

TL;DR

This paper demonstrates that zeros of the Riemann zeta function induce significant oscillations in the error term of the prime number theorem, with results nearly optimal in magnitude and localization.

Contribution

It establishes a close connection between zeros of the zeta function and large oscillations in the prime number theorem's error term, refining understanding of their relationship.

Findings

Zeros of the zeta function cause large oscillations in the error term.

Results are nearly optimal in magnitude and localization of these oscillations.

Abstract

Let $\sigma+i\gamma$ be a zero of the Riemann zeta function to the right of the line $\frac{1}{2}+it$. We show that this zero causes large oscillations of the error term of the prime number theorem. Our result is close to optimal both in terms of the magnitude and in the localization of large values for the error term.