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Applied Physics and Mathematics

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PROBABILISTIC APPROACH TO THE DISTRIBUTION OF PRIMES IN THE NATURAL NUMBERS AND ARITHMETIC PROGRESSION
V.L. VOL`FSON

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 affi rming 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) = 1also built in this paper. The author proved the probability
estimates for the deviation R(x,k,1) = |π(x,k,l) – Li(x)/φ(k)|. The author has analyzed the probability models of the distribution of primes in the arithmetic progression, and affi rming the validity of probabilistic estimates of proved deviations R(x,k,1) stronger, than the estimates made under the assumption of
the extended Riemann conjecture. Elliott-Halberstam conjecture Σ1≤k≤xa{max(k,l)=1R(x,k,1)}≤Cx/lnA(x) was proved in this paper with probability arbitrarily close to 1 for all 0 Keywords: probabilistic model, limit distribution of the random variable, normal distribution, probability estimates, prime numbers, Riemann conjecture, Legendre’s conjecture, Cramer’s conjecture, Elliott-Halberstam conjecture.

Contacts: E-mail: znakvicvolf@mail.ru

Pp. 083-093.

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