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

This work is a probabilistic study of the ‘primes’  of the Cramér model, which consists with sums \(S_n =\sum _{i=3}^n \xi _i\) $S_n={\sum }_{i=3}^n{\xi }_i$ , \(n\ge 3\) $n\ge 3$ , where \(\xi _i\) ${\xi }_i$ are independent random variables such that \(\mathbb {P}\{\xi _i= 1\}= 1-\mathbb {P}\{\xi _i= 1\}=1/{\log i}\) $\mathbb{P}\{{\xi }_i=1\}=1-\mathbb{P}\{{\xi }_i=1\}=1/logi$ , \(i\ge 3\) $i\ge 3$ . We prove that there exists a set of integers \(\mathscr {S} \) $\mathcal{S}$ of density 1 such that 0.0.1 \(\begin{aligned} \liminf _{ \mathscr {S}\ni n\rightarrow \infty } (\log n)\mathbb {P}\{S_n\ \hbox {prime} \} \ge \frac{1}{\sqrt{2\pi e}\, } , \end{aligned}\) $$\begin{array}{c}\hfill \underset{\mathcal{S}\ni n\to \infty }{lim\; inf}(logn)\mathbb{P}\{S_n\text{prime}\}\ge \frac{1}{\sqrt{2\pi e}},\end{array}$$ and that for \(b>\frac{1}{2}\) $b>\frac{1}{2}$ , the formula 0.0.2 \(\begin{aligned} \mathbb {P}\{S_n\ \text {prime}\, \} \, =\, \frac{ (1+ o( 1) )}{ \sqrt{2\pi B_n } } \int _{m_n-\sqrt{ 2bB_n\log n}}^{m_n+\sqrt{ 2bB_n\log n}} \, e^{- \frac{(t- m_n)^2}{ 2 B_n } }\, \textrm{d}\pi (t), \end{aligned}\) $$\begin{array}{c}\hfill \mathbb{P}\{S_n\text{prime}\}=\frac{(1+o(1))}{\sqrt{2\pi B_n}}{\int }_{m_n-\sqrt{2b{B}_{n}logn}}^{m_n+\sqrt{2b{B}_{n}logn}}{e}^{-\frac{{(t-m_n)}^2}{2B_n}}\text{d}\pi (t),\end{array}$$ in which \(m_n={\mathbb E \,}S_n,B_n=\textrm{Var }\,S_n\) $m_n=\mathbb{E}{S}_n,B_n=\text{Var}{S}_n$ , holds true for all \(n\in \mathscr {S}\) $n\in \mathcal{S}$ , \(n\rightarrow \infty \) $n\to \infty$ . Further we prove that for any \(0<\eta <1\) $0<\eta <1$ , and all n large enough and \( \zeta _0\le \zeta \le \exp \big \{ \frac{c\log n}{\log \log n}\big \}\) ${\zeta }_0\le \zeta \le exp\{\frac{clogn}{loglogn}\}$ , letting \(S'_n= \sum _{j= 8}^n \xi _j\) $S_n^{\prime }={\sum }_{j=8}^n{\xi }_j$ , \(\begin{aligned} \mathbb {P}\big \{ S'_n \zeta \hbox {-quasiprime}\big \} \,\ge \, (1-\eta ) \frac{ e^{-\gamma } }{ \log \zeta }, \end{aligned}\) $$\begin{array}{c}\hfill \mathbb{P}\{S_n^{\prime }\zeta \text{-quasiprime}\}\ge (1-\eta )\frac{e^{-\gamma }}{log\zeta },\end{array}$$ according to Pintz’s terminology, where \(c>0\) $c>0$ and \(\gamma \) $\gamma$ 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 occurences of intervals I such as for some \(z>0\) $z>0$ , 0.0.3 \(\begin{aligned} \sup _{n\in I} \frac{|S_n-m_n|}{ \sqrt{B_n}}\le z, \end{aligned}\) $$\begin{array}{c}\hfill \underset{n\in I}{sup}\frac{|S_n-m_n|}{\sqrt{B_n}}\le z,\end{array}$$ which are tied with the spectrum of the Sturm-Liouville equation.