arXiv subjects

Victor Volfson

Publications and source records attributed to Victor Volfson.

40 records · Page 3Linked to original sources

arXiv2015

The distribution of prime numbers and tuples in the natural numbers

We studied two probabilistic models of the distribution of primes in the natural number [1].The paper considers the third probabilistic model of the distribution of primes in the natural number. The author proved that the results obtained by considering three probabilistic models are similar. It is shown that the accuracy of these probabilistic models exceeds the Riemann hypothesis with a certain probability. The author constructed probabilistic models for the analysis of the average distance between neighboring prime numbers and k-tuples and found probability distributions of these random variables. The estimates of these random variables are obtained based on these probabilistic models. The resulting estimates tested on a large volume of statistical data.

math.NT↗

arXiv2015

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

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

math.GM↗

arXiv2014

Quantitative indicators of the solutions of Diophantine equations and systems in the domain of the natural numbers

The paper shows that the asymptotic density of solutions of Diophantine equations or systems of the natural numbers is 0. The author provides estimation methods and estimates number, density and probability of k- tuples $ $ to be the solution of the algebraic equations of the first, second and higher orders in two or more variables, non-algebraic Diophantine equations and systems of Diophantine equations in the domain of the natural numbers. The estimate for the number of positive integer solutions of the second-order Diophantine equations in two, three or more variables is geometrically proved in the paper. The author proves the assertion about the number of solutions of algebraic Diophantine equations of higher orders in the domain of the natural numbers. The author provides the estimates for the asymptotic behavior of quantitative solutions of Diophantine equations and systems in the domain of the natural numbers.

math.NT↗

arXiv2014

Contradictions in some primes conjectures

This paper demonstrates that from the Cramer's, Hardy-Littlewood's and Bateman-Horn's conjectures (suggest that the probability of a large positive integer being $x$ a prime - $\frac {1} {\ln(x)}$) it follows that the events consisting in a positive integer $x$ being not divisible by different primes are dependent with the ratio $0,5e^γ$ ($γ$ - Euler's constant). In establishing Hardy-Littlewood's and Bateman-Horn's conjectures, their authors followed the first suggestion by another one assuming the independence of the above-mentioned events, which on the basis of the first suggestion and Merten's theorem is not exactly. This paper demonstrates why these suggestions do not lead to an erroneous result using the Hardy-Littlewood's conjecture for twin primes as an example. The author provides generalized conjectures, which if taken together with the first suggestion, make Hardy-Littlewood's prime $k$ - tuples and Bateman-Horn's conjectures true.

math.NT↗