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

2015/01/28 by Volfson, Victor
#11N05 #FOS: Mathematics #General Mathematics (math.GM)

paper · doi:10.48550/arxiv.1501.07267

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)=|π(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)=|π(x,k, l)-Li(x)/φ(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 ∑_{1 ≤ k ≤ x^a} max_(k,l)=1[R(x,k,l)] ≤ C x/ln^A(x) was proved in this paper with probability arbitrarily close to 1 for all 00, based on the probability estimates.

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• The distribution of prime numbers and tuples in the natural numbers
• Determination of the asymptotic behavior of probabilistic characteristics of arithmetic functions and some other questions of probabilistic number theory
• Algorithm, probability, and prime numbers
• The distribution of the variance of primes in arithmetic progressions
• New conjectures in number theory - The distribution of prime numbers