The distribution of prime numbers and tuples in the natural numbers

TL;DR

This paper explores three probabilistic models of prime distribution, demonstrating their similarity and showing they can predict prime patterns with accuracy surpassing the Riemann hypothesis in certain probabilistic terms.

Contribution

It introduces a third probabilistic model for prime distribution, compares it with existing models, and provides new estimates for prime gaps and tuples based on these models.

Findings

Probabilistic models are similar in predicting prime distribution.

Models outperform the Riemann hypothesis in certain probabilistic measures.

New probability distributions for prime gaps and k-tuples are derived and tested.

Abstract

We studied two probabilistic models of the distribution of primes in the natural number [1].The paper considers the third probabilistic model of the distribution of primes in the natural number. The author proved that the results obtained by considering three probabilistic models are similar. It is shown that the accuracy of these probabilistic models exceeds the Riemann hypothesis with a certain probability. The author constructed probabilistic models for the analysis of the average distance between neighboring prime numbers and k-tuples and found probability distributions of these random variables. The estimates of these random variables are obtained based on these probabilistic models. The resulting estimates tested on a large volume of statistical data.