Discrete dynamics of Eratosthenes sieve

Source abstract

We study Eratosthenes sieve as a discrete dynamic system. At each stage of the sieve there is a cycle of gaps ${\mathcal G}(p^\#)$ of length $φ(p^\#)$ and span $p^\#$. There is a recursion ${\mathcal G}(p_k^\#)\longrightarrow {\mathcal G}(p_{k+1}^\#)$ that creates the next cycle from the current one. If we take initial conditions from the cycle ${\mathcal G}(p_0^\#)$, then for all constellations of span $|s| < 2p_1$, including gaps $g < 2p_1$, the driving terms of various lengths form Markov chains. These yield {\it exact} models for the populations $n_s(p_k^\#)$ for all further stages of the sieve. If $s$ is an admissible constellation of length $J$, then its population $n_{s,J}(p^\#)$ grows as $Θ\left( \prod (q-J-1)\right)$. So we factor out the superexponential growth to obtain the exact model for the relative population $w_s(p_k^\#)$ of the constellation $s$ across all further stages of the sieve. $$ w_{s,J}(p_k^\#) \; = \; n_{s,J}(p_k^\#) \, / \, \prod_{J+1 < p \le p_k} (p-J-1) $$ The asymptotic value of the relative population is a constant ${w_{s,J}(\infty) \ge 1}$ that depends only on the odd prime factors that divide a span in $s$. Assuming that the instances of a constellation $s$ are approximately uniformly distributed in ${\mathcal G}(p_k^\#)$, we develop first-order estimates of the number of instances $s$ that would occur in the interval of survival $ΔH(p_k) = (p_k^2, p_{k+1}^2]$. We define a statistic $η_s(p_k)$, the quadratic density of the constellation $s$ over the interval $ΔH(p_k)$. We show that the first-order estimates $\widehat{η_g}(p)$ for prime gaps agree with samples up to $5.677\,E14$.