A probabilistic approach to the prime numbers distribution

by @ulaulaman about #prime_numbers #arXiv #mathematics

The prime numbers theorem states the asymptotic approximation for the prime-counting function. The first statement for the theorem is given by Euler in 1737 (pdf):

The sum of the series of the reciprocals of the prime numbers, \[\frac{1}{2} + \frac{1}{3} + \frac{1}{5} + \frac{1}{7} + \frac{1}{11} + \frac{1}{13} + \cdots\] is infinitely large, but it is infinitely many times less than the sum of the harmonic series, \[1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \frac{1}{5} + \cdots\] Furthemore, the sum of the former series is like the logarithm of the sum of the latter series.

After Euler, the most known attempt to evaluate the prime numbers distribution is given by Bernard Riemann, and the most recent is posted on arXiv some days ago:

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 $\sum_{1 \leq k \leq x^a} \max_{(k,l) = 1} [R(x,k,l)] \leq Cx/\ln^A(x)$ was proved in this paper with probability arbitrarily close to 1 for all $0 < a < 1$ and $A > 0$, based on the probability estimates.

Victor Volfson (2015). Probabilistic approach to the distribution of primes and to the proof of Legendre and Elliott-Halberstam conjectures, arXiv: http://arxiv.org/abs/1501.07267v1