Prime numbers in logarithmic intervals

TL;DR

This paper introduces new estimates for the distribution of primes in short intervals and applies these results to analyze prime gaps, intervals containing primes, and the impact of conjectures like the Riemann Hypothesis on prime distributions.

Contribution

It provides novel bounds for moments of primes in short intervals and extends results on prime gaps and prime-containing intervals, under both unconditional and conditional assumptions.

Findings

Proves a positive proportion of primes have intervals of length \\lambda \\log X containing at least one prime for \\lambda > 1/2.

Improves previous results on the size of \\lambda > 1 with positive proportion of prime-free intervals.

Establishes new theorems on prime gaps and prime distribution in short intervals, both unconditionally and assuming the Riemann Hypothesis.

Abstract

Let $X$ be a large parameter. We will first give a new estimate for the integral moments of primes in short intervals of the type $(p,p+h]$, where $p\leq X$ is a prime number and $h=\odi{X}$. Then we will apply this to prove that for every $\lambda>1/2$ there exists a positive proportion of primes $p\leq X$ such that the interval $(p,p+ \lambda\log X]$ contains at least a prime number. As a consequence we improve Cheer and Goldston's result on the size of real numbers $\lambda>1$ with the property that there is a positive proportion of integers $m\leq X$ such that the interval $(m,m+ \lambda\log X]$ contains no primes. We also prove other results concerning the moments of the gaps between consecutive primes and about the positive proportion of integers $m\leq X$ such that the interval $(m,m+ \lambda\log X]$ contains at least a prime number. The last application of these techniques are two theorems (the first one unconditional and the second one in which we assume the validity of the Riemann Hypothesis and of a form of the Montgomery pair correlation conjecture) on the positive proportion of primes $p\leq X$ such that the interval $(p,p+ \lambda\log X]$ contains no primes.