Critical probabilistic characteristics of the Cramér model for primes and arithmetical properties

Abstract

This work is a probabilistic study of the 'primes' of the Cramér model. We prove that there exists a set of integers $\mathcal S$ of density 1 such that \begin{equation}\liminf_{ \mathcal S\ni n\to\infty} (\log n)\mathbb{P} \{S_n\ \hbox{prime} \} \ge \frac{1}{\sqrt{2πe}\, }, \end{equation} and that for $b>\frac12$, the formula \begin{equation} \mathbb{P} \{S_n\ \text{prime}\, \} \, =\, \frac{ (1+ o( 1) )}{ \sqrt{2πB_n } } \int_{m_n-\sqrt{ 2bB_n\log n}}^{m_n+\sqrt{ 2bB_n\log n}} \, e^{-\frac{(t - m_n)^2}{ 2 B_n } }\, {\rm d}π(t), \end{equation} in which $m_n=\mathbb{E} S_n,B_n={\rm Var }\,S_n$, holds true for all $n\in \mathcal S$, $n\to \infty$. Further we prove that for any $0<η<1$, and all $n$ large enough and $ ζ_0\le ζ\le \exp\big\{ \frac{c\log n}{\log\log n}\big\}$, letting $S'_n= \sum_{j= 8}^n ξ_j$, \begin{eqnarray*} \mathbb{P}\big\{ S'_n\hbox{\ $ζ$-quasiprime}\big\} \,\ge \, (1-η) \frac{ e^{-γ} }{ \log ζ}, \end{eqnarray*} according to Pintz's terminology, where $c>0$ and $γ$ is Euler's constant. We also test which infinite sequences of primes are ultimately avoided by the 'primes' of the Cramér model, with probability 1. Moreover we show that the Cramér model has incidences on the Prime Number Theorem, since it predicts that the error term is sensitive to subsequences. We obtain sharp results on the length and the number of occurrences of intervals $I$ such as for some $z>0$, \begin{equation}\sup_{n\in I} \frac{|S_n-m_n|}{ \sqrt{B_n}}\le z, \end{equation} which are tied with the spectrum of the Sturm-Liouville equation.