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 $|\pi (x)-\text{li}(x)|=O(\sqrt{\text{li}(x)})$. 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.

Minor revisions in preparation for submission