  This work is a probabilistic study of the 'primes' of the Cram\'er 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 $0<\eta<1$, and all $n$ large enough and $ \zeta_0\le \zeta\le \exp\big\{
\frac{c\log n}{\log\log n}\big\}$, letting $S'_n= \sum_{j= 8}^n \xi_j$,
\begin{eqnarray*} \mathbb{P}\big\{ S'_n\hbox{\ $\zeta$-quasiprime}\big\} \,\ge
\, (1-\eta) \frac{ e^{-\gamma} }{ \log \zeta }, \end{eqnarray*} according to
Pintz's terminology, where $c>0$ and $\gamma$ is Euler's constant. We also test
which infinite sequences of primes are ultimately avoided by the 'primes' of
the Cram\'er model, with probability 1. Moreover we show that the Cram\'er
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.

Number Theory