AI 中文总结
研究埃拉托斯特尼筛法的离散动力学，通过递归与马尔可夫链建立星群种群的精确模型，推导相对种群公式，证明素间隙一阶估计与大样本一致。
AI 中文摘要
我们将埃拉托斯特尼筛法作为离散动力系统进行研究。在筛法的每个阶段，存在一个长度为φ(p#)、跨度为p#的间隙循环G(p#)，存在递归关系G(p_k#)→G(p_{k+1}#)，该递归从当前循环生成下一个循环。若取初始条件来自循环G(p_0#)，则对于所有跨度|s|<2p_1的星群（包括间隙g<2p_1），不同长度的驱动项构成马尔可夫链，这些马尔可夫链为筛法后续所有阶段的种群n_s(p_k#)提供精确模型。若s是长度为J的可容许星群，其种群n_{s,J}(p#)呈Θ(∏(q-J-1))增长，因此我们分解超指数增长，得到星群s在筛法后续所有阶段的相对种群w_s(p_k#)的精确模型，公式为：w_{s,J}(p_k#)=n_{s,J}(p_k#)/∏_{J+1<p≤p_k}(p-J-1)。相对种群的渐近值是仅取决于s中跨度所分奇素因子的常数w_{s,J}(∞)≥1。假设星群s的实例在G(p_k#)中近似均匀分布，我们开发了对区间ΔH(p_k)=(p_k²,p_{k+1}²]中出现的星群s实例数量的一阶估计。我们定义统计量η_s(p_k)，即星群s在区间ΔH(p_k)上的二次密度，证明了素间隙的一阶估计η̂_g(p)与高达5.677E14的样本一致。
英文摘要
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 $5.677\,E14$.
Comments24 pages, 13 figures
