CRITICAL PROBABILISTIC CHARACTERISTICS OF THE CRAMÉR MODEL FOR PRIMES AND ARITHMETICAL PROPERTIES

This work is a probabilistic study of the 'primes' of the Cramer 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{2\pi e}\, }, \end{equation} and that for $b>\frac12$, the formula \begin{equation} \mathbb{P} \{S_n\ \text{prime}\, \} \, =\, \frac{ (1+ o( 1) )}{ \sqrt{2\pi B_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}\pi(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 $0 0$ and $\gamma$ is Euler's constant. We also test which infinite sequences of primes are ultimately avoided by the 'primes' of the Cramer model, with probability 1. Moreover we show that the Cramer 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.