On the connection between zero-free regions and the error term in the Prime Number Theorem

TL;DR

This paper establishes a new upper bound for the error term in the Prime Number Theorem based on zero-free regions, extending previous results to Beurling zeta functions and demonstrating near-sharpness through oscillation analysis.

Contribution

It introduces a generalized method linking zero-free regions to error bounds, applicable to Beurling zeta functions, and constructs functions with zeros on prescribed contours.

Findings

Refined upper bounds for the error term in the Prime Number Theorem.

Construction of Beurling zeta functions with zeros on specific contours.

Demonstration of oscillation behavior indicating near-sharpness of bounds.

Abstract

We provide for a wide class of zero-free regions an upper bound for the error term in the Prime Number Theorem, refining works of Pintz (1980), Johnston (2024), and R\'ev\'esz (2024). Our method does not only apply to the Riemann zeta function, but to general Beurling zeta functions. Next we construct Beurling zeta functions having infinitely many zeros on a prescribed contour, and none to the right, for a wide class of such contours. We also deduce an oscillation result for the corresponding error term in the Prime Number Theorem, showing that our aforementioned refinement is close to being sharp.