Probabilistic approach to the distribution of primes and to the proof of Legendre and Elliott-Halberstam conjectures

TL;DR

This paper develops probabilistic models for prime distribution, proving key conjectures like Legendre and Elliott-Halberstam with high probability, surpassing classical assumptions such as the Riemann hypothesis.

Contribution

The paper introduces new probabilistic estimates for prime distribution deviations and proves significant conjectures with high probability, advancing understanding beyond traditional deterministic methods.

Findings

Proved Legendre's conjecture with probability close to 1.

Established probabilistic estimates for prime deviations in arithmetic progressions.

Proved the Elliott-Halberstam conjecture with high probability for certain parameters.

Abstract

Probabilistic models for the distribution of primes in the natural numbers are constructed in the article. The author found and proved the probabilistic estimates of the deviation $R(x)=|\pi(x)- Li(x)|$. The author has analyzed the probabilistic models of the distribution of primes in the natural numbers and affirmed the validity of the probabilistic estimates of proved deviations $R(x)$ stronger than the estimates made under the assumption of Riemann conjecture. Legendre's conjecture was proved in this paper with probability arbitrarily close to 1 based on the probability estimates. Probabilistic models for the distribution of primes in the arithmetic progression $ki+l, (k,l)=1$ are also built in this paper. The author has proved the probability estimates for the deviation $R(x,k,l)=|\pi(x,k, l)-Li(x)/\varphi(k)|$. He has analyzed the probability models of the distribution of primes in the arithmetic progression and affirmed the validity of probabilistic estimates of proved deviations $R(x,k,l)$ stronger than the estimates made under the assumption of the extended Riemann conjecture. Elliott-Halberstam conjecture $\sum_{1 \leq k \leq x^a} {\max_{(k,l)=1}[R(x,k,l)]} \leq C x/\ln^A(x)$ was proved in this paper with probability arbitrarily close to 1 for all $0<a<1$ and $A>0$, based on the probability estimates.