Critical probabilistic characteristics of the Cram\'er model for primes and arithmetical properties

TL;DR

This paper probabilistically analyzes the Cramér model for primes, establishing density, asymptotic formulas, and properties of prime-like sequences, revealing insights into prime distribution and model predictions.

Contribution

It provides new probabilistic results on the distribution and properties of primes within the Cramér model, including asymptotic formulas and incidence results.

Findings

Existence of a density 1 set with lower bounds on prime probabilities

Asymptotic integral formula for prime probabilities in the model

Predictions on the length and frequency of prime-like intervals

Abstract

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.