Primes in the intervals between primes squared

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.