Half-isolated zeros and zero-density estimates

TL;DR

This paper introduces a novel method for detecting zeros of the Riemann zeta function, demonstrating the existence of few 'half-isolated' zeros and improving zero-density estimates, with implications for primes in short intervals.

Contribution

The paper presents a new zero detection method sensitive to zero distribution and enhances zero-density bounds assuming zeros lie on finitely many vertical lines.

Findings

Few 'half-isolated' zeros exist.

Improved zero-density estimates under specific assumptions.

Implications for primes in short intervals.

Abstract

We introduce a new method to detect the zeros of the Riemann zeta function which is sensitive to the vertical distribution of the zeros. This allows us to prove there are few `half-isolated' zeros. By combining this with classical methods, we improve the Ingham-Huxley zero-density estimate under the assumption that the non-trivial zeros of the zeta function are restricted to lie on a finite number of fixed vertical lines. This has new consequences for primes in short intervals under the same assumption.