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

Fred B. Holt

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 $ΔH(p_k)=[p_k^2, p_{k+1}^2]$ 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 $ΔH(p_k)$ 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 $η_s(p_k)$, 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.

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 ) in the cycle of gaps at each stage of the sieve. The gaps in the interval are fixed in 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 have population distributions that reflect the relative population models . 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 , which is the expected population of the constellation in the intervals for . We show that once a gap occurs in , its expected quadratic density increases across all subsequent stages of the sieve. Regarding Legendre's conjecture, beyond postulating one prime in the interval , the quadratic density predicts the populations of several prime gaps within this interval.

Paper Structure

This paper contains 10 sections, 4 theorems, 35 equations, 11 figures.

Table of Contents

  1. Evolution of populations of constellations in ${\mathcal{G}}({p}^{\#})$.
  2. Estimating survival through the sieve
  3. The quadratic density of constellations
  4. In the large, quadratic density $\eta$ is non-decreasing.
  5. Regarding $g_3/p_k$.
  6. Regarding $\delta_s$.
  7. Samples of short constellations.
  8. Discussion of expected survival
  9. Regarding Legendre's Conjecture.
  10. Future directions

Key Result

Lemma 3.1

Let $s$ be a constellation in ${\mathcal{G}}({p_k}^{\#})$ of length $J < p_{k+1}-2$ and span $|s| < 2p_{k+1}$. Then

Figures (11)

  • Figure 1: We have previously established the exact models for the populations $n_s({p_k}^{\#})$ and relative populations $w_s({p_k}^{\#})$ in the cycle of gaps ${\mathcal{G}}({p_k}^{\#})$, for all constellations $s$ with span $|s| \le 2p_1$. Shown here are the relative population models $w_{g,1}({p_k}^{\#})$ for $p_0=37$ and all gaps $g \le 82$. The graphs start at the right, where $p_0=37$ and the system parameter $\lambda=1$. As the sieve continues, $\lambda \longrightarrow 0$ as $p \longrightarrow \infty$, and we follow the graphs to the left. As the sieve proceeds through enormous primes, the graphs converge toward their asymptotic values at $\lambda=0$.
  • Figure 2: Shown here are samples of the populations of gaps within intervals of survival $\Delta H(p)$ for four different values of $\lambda$. The first-order estimates based on $w_g(\lambda)$ and a $2^{\rm nd}$-order correction are represented by the piecewise lines. The samples from the individual $\Delta H(p)$ are stacked and color-coded by the gap ${g=p_{k+1}-p_k}$.
  • Figure 3: These graphs are statistical summaries of samples of quadratic density $\eta$ corresponding to the first two panels in Figure \ref{['DelHFig']}. Shown are the samples of the quadratic densities themselves, as individual markers. A black diamond marks the mean for the sample, and the short lines mark one and two standard deviations of the sample.
  • Figure 4: The model for the population of a constellation $s$ of length $J$ when the span $|s| < 2p_1$, illustrated as a Markov chain across its driving terms.
  • Figure 5: The model for the relative population $w_s(p_k^\#)$ of a constellation $s$ of length $J$ and span $|s| < 2p_1$, illustrated as a Markov chain.
  • ...and 6 more figures

Theorems & Definitions (9)

  • Conjecture 1.1
  • Conjecture 2.1
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Theorem 3.3
  • proof
  • Lemma 3.4