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

TL;DR

This paper models the Eratosthenes sieve as a dynamic system, introduces quadratic density to estimate prime gap populations, and explores implications for Legendre's conjecture.

Contribution

It advances the understanding of prime gap survival in the sieve and introduces quadratic density as a new predictive measure related to Legendre's conjecture.

Findings

Quadratic density increases after a gap occurs in the sieve.

Population distributions of prime gaps reflect relative models over survival intervals.

The quadratic density predicts multiple prime gaps within square intervals.

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.