Short intervals with a given number of primes

TL;DR

This paper investigates the distribution of prime numbers within short intervals scaled by a factor of  log n, providing lower bounds on the count of such intervals containing a fixed number of primes, supporting a classical conjecture.

Contribution

It establishes a lower bound of at least x^{1 - o(1)} for the count of integers n where the interval (n, n +  log n] contains exactly m primes, advancing understanding of prime distribution in short intervals.

Findings

Number of such n is at least x^{1 - o(1)}.

Supports the conjecture on the asymptotic distribution of primes in short intervals.

Provides quantitative bounds related to prime counts in scaled intervals.

Abstract

A well-known conjecture asserts that, for any given positive real number $\lambda$ and nonnegative integer $m$, the proportion of positive integers $n \le x$ for which the interval $(n,n + \lambda\log n]$ contains exactly $m$ primes is asymptotically equal to $\lambda^me^{-\lambda}/m!$ as $x$ tends to infinity. We show that the number of such $n$ is at least $x^{1 - o(1)}$.