Some Results on a Conjecture of Hardy and Littlewood

TL;DR

This paper investigates a conjecture by Hardy and Littlewood regarding the distribution of primes in specific intervals, providing new verified ranges using explicit estimates of the prime counting function.

Contribution

It offers new explicit bounds and ranges where the Hardy-Littlewood conjecture on prime counts in intervals is confirmed.

Findings

Confirmed the conjecture for new ranges of m and n

Provided explicit estimates for the prime counting function

Extended the validity of the conjecture based on numerical bounds

Abstract

Let $m$ and $n$ be positive integers with $m,n \geq 2$. The second Hardy-Littlewood conjecture states that the number of primes in the interval $(m,m+n]$ is always less than or equal to the number of primes in the interval $[2,n]$. Based on new explicit estimates for the prime counting function $\pi(x)$, we give some new ranges in which this conjecture holds.