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

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.