Primes in the intervals between primes squared

TL;DR

This paper investigates the distribution of primes within intervals between consecutive prime squares, proposing conjectures about prime counts, variance, and their relation to the Riemann hypothesis, based on the unique sieving property of these intervals.

Contribution

It introduces new conjectures on prime distribution in prime square intervals and models the prime counting function as a sum of correlated random variables.

Findings

Evidence supporting the conjecture that $\pi_k \\sim |s_k|/ \\log p_{k+1}^2$

Proposal that $\\pi(x)$ can be modeled as a sum of correlated variables

Conjecture that $|\\pi(x)-li(x)|=O(\\sqrt{li(x)})$

Abstract

The set of short intervals between consecutive primes squared has the pleasant---but seemingly unexploited---property that each interval $s_k:=\{p_k^2, \dots,p_{k+1}^2-1\}$ is fully sieved by the $k$ first primes. Here we take advantage of this essential characteristic and present evidence for the conjecture that $\pi_k \sim |s_k|/ \log p_{k+1}^2$, where $\pi_k$ is the number of primes in $s_k$; or even stricter, that $y=x^{1/2}$ is both necessary and sufficient for the prime number theorem to be valid in intervals of length $y$. In addition, we propose and substantiate that the prime counting function $\pi(x)$ is best understood as a sum of correlated random variables $\pi_k$. Under this assumption, we derive the theoretical variance of $\pi(p_{k+1}^2)=\sum_{j=1}^k \pi_j$, from which we are led to conjecture that $|\pi({x})-\textrm{li}(x)| =O(\sqrt{\textrm{li}(x)})$. Emerging from our investigations is the view that the intervals between consecutive primes squared hold the key to a furthered understanding of the distribution of primes; as evidenced, this perspective also builds strong support in favour of the Riemann hypothesis.