Surviving Eratosthenes sieve I: quadratic density and Legendre's conjecture

Abstract

We have been studying Eratosthenes sieve as a discrete dynamic system, obtaining exact models for the relative populations for small gaps (currently gaps $g \le 82$) in the cycle of gaps ${\mathcal G}(p^\#)$ at each stage of the sieve. The gaps in the interval $\Delta H(p_k)=[p_k^2, p_{k+1}^2]$ are fixed in ${\mathcal G}(p^\#)$ and survive all subsequent stages of the sieve to be confirmed as gaps between primes. We have shown that samples of gaps between primes over these intervals of survival $\Delta H(p_k)$ have population distributions that reflect the relative population models $w_g(p_k^\#)$. This paper advances our study of the estimates of survival across stages of the sieve. Inspired by Legendre's conjecture, we introduce the concept of quadratic density $\eta_s(p_k)$, which is the expected population of the constellation $s$ in the intervals $[n^2, (n+1)^2]$ for $p_k \le n < p_{k+1}$. We show that once a gap occurs in ${\mathcal G}(p^\#)$, its expected quadratic density increases across all subsequent stages of the sieve. Regarding Legendre's conjecture, beyond postulating one prime in the interval $[n^2,(n+1)^2]$, the quadratic density predicts the populations of several prime gaps within this interval.