A note on zero-density approaches for the difference between consecutive primes

TL;DR

This paper generalizes key zero-density results relating to primes in short intervals, clarifying their connections and refining bounds under the Density Hypothesis, thus advancing understanding of prime distribution in small ranges.

Contribution

It extends Ingham's theorem and related zero-density estimates to more general settings, improving bounds for primes in short intervals under the Density Hypothesis.

Findings

Generalized Ingham's theorem under the Density Hypothesis

Refined bounds for primes in short intervals

Clarified links between zero-free regions and prime distribution

Abstract

In this note, we generalise two results on prime numbers in short intervals. The first result is Ingham's theorem which connects the zero-density estimates with short intervals where the prime number theorem holds, and the second result is due to Heath-Brown and Iwaniec, which derives the weighted zero-density estimates used for obtaining the lower bound for the number of primes in short intervals. The generalised versions of these results make the connections between the zero-free regions, zero-density estimates, and the primes in short intervals more transparent. As an example, the generalisation of Ingham's theorem implies that, under the Density Hypothesis, the prime number theorem holds in $[x - \sqrt{x}\exp(\log^{2/3+\varepsilon}x), x]$, which refines upon the classic interval $[x - x^{1/2+ \varepsilon}, x]$.