arXiv subjects

Victor Volfson

Publications and source records attributed to Victor Volfson.

At least 37 records · Page 2Linked to original sources

arXiv2021

Asymptotics of probability characteristics of additive arithmetic functions

We study the questions of determining the asymptotics of the probabilistic characteristics of additive arithmetic functions in the paper, regardless of whether they have a limit distribution or not. Several assertions are proved about the estimation of the asymptotics of the probabilistic characteristics of strongly additive arithmetic functions, as well as additive functions of the class that have the same asymptotic behavior of the probabilistic characteristics, as for strongly additive arithmetic functions.

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arXiv2020

Determination of the asymptotic behavior of probabilistic characteristics of arithmetic functions and some other questions of probabilistic number theory

One of the questions of distribution of prime numbers is considered in the article. It is shown what error is obtained from the assumption that the asymptotic density of a sequence of primes is a probability. Various forms of an analogue of the law of large numbers for arithmetic functions and, in particular, the Hardy-Ramunajan theorem are obtained. A method is given for finding asymptotics of the probabilistic characteristics of arithmetic functions.

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arXiv2020

Asymptotics for sums of functions of primes

This work gives a general approach to the determination of the asymptotic behavior of the sums of functions of primes based on the distribution of primes. It refines the estimate of the remainder term of the asymptotic expansion of the sums of functions of primes. Also, the necessary and sufficient conditions for the existence of these asymptotics are proved in the paper.

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arXiv2020

Asymptotic of summation functions

We will study the asymptotic behavior of summation functions of a natural argument, including the asymptotic behavior of summation functions of a prime argument in the paper. A general formula is obtained for determining the asymptotic behavior of the sums of functions of a prime argument based on the asymptotic law of primes. We will show, that under certain conditions: $\sum_{p \leq n} {f(p)}= \sum_{k=2}^n {\frac {f(k)}{\log(k)}(1+o(1))}$, where $p$ is a prime number. In the paper, the necessary and sufficient conditions for the fulfillment of this formula are proved.

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arXiv2020

Asymptotic of the greatest distance between adjacent primes and the Hardy-Littlewood conjecture

The paper substantiates the conjecture of the asymptotic behavior of the largest distance between consecutive primes: $sup_{p_i \leq x}(p_{i+1}-p_i) \sim 2e^{-γ} \log^2(x)$, where $γ$ is the Euler constant. The Hardy-Littlewood conjecture about the number of prime tuples is investigated and the rationale for this conjecture is given, taking into account the fact that a large natural number is not divisible by primes. It also substantiates why the accuracy of this conjecture is not affected by another assumption about the probability of a natural number being prime, although such a probability does not exist. The paper also considers the distribution of prime tuples using a mathematical model based on the Hardy-Littlewood conjecture.

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arXiv2020

Determination of the asymptotic behavior of the number of natural solutions for certain types of diagonal Diophantine equations

We obtain asymptotic upper bounds for the number of natural solutions of the following diagonal Diophantine equations in a hypercube with side - $N$ in the paper: $x_1 = x_2^k+...+x_s^k$, $x_1^k = x_2^k+...+x_s^k$, $x_1 = \sum_{j=2}^s {x_j}^{k_j}$, where $k,s,k_j$ are the natural numbers.

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arXiv2019

Summation arithmetic functions with bounded terms, having a limit normal distribution law

The paper considers the properties of pseudo stationarity in a broad sense and pseudo strong mixing for sequences of random variables corresponding to arithmetic functions. Assertions on this topic have been proven. The implementation of these properties for known arithmetic functions has been verified. The article proves a statement about sufficient conditions under which a summation arithmetic function with bounded terms has a limit normal distribution law. The fulfillment of the specified sufficient conditions for known arithmetic functions is considered.

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arXiv2019

Asymptotic of summation arithmetic functions, the limit for which is the law of normal distribution

Summation arithmetic functions with asymptotically independent terms are studied in the paper, the limit of which is the law of normal distribution. Assertions about the asymptotic behavior of the indicated functions are proved.

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arXiv2019

Summation arithmetic functions with asymptotically independent summands

The summation arithmetic functions with asymptotically independent summands are studied in the paper. We prove statements about the condition under which the summation arithmetic functions have asymptotically independent summands. It is also prove that the limiting distribution of the summation arithmetic function with asymptotically independent summands is normal under certain conditions for summands of the summation arithmetic function.

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arXiv2018

Investigations of Mertens and Liouville summation functions

Summation arithmetic functions of Mertens and Liouville are investigated in the paper. It is proved that the limiting distribution of these functions is the normal. It is also shown that the estimating of standard deviation of these functions $O(n^{1/2})$ cannot be improved. The estimate of the average of Liouville summation function is found. The estimate of the order of growth of the ratio of the summation functions of Mertens and Liouville is also found.

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arXiv2018

Comparison of probabilistic and exact methods for estimating the asymptotic behavior of summation arithmetic functions

The paper compares probabilistic and exact methods for estimating the asymptotic behavior of summation arithmetic functions, and estimates of the results are obtained by precise methods. Conditions for stationarity in the broad sense are investigated for summation arithmetic functions. A lemma and theorems about the estimation of the standard deviation for the summation arithmetic Mertens and Lowville functions completely satisfying the stationarity conditions in the broad sense are proved.

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arXiv2018

Investigations of the limit distribution and the asymptotic behavior of summation arithmetic functions

The aim of the paper is to study the limit distributions and the asymptotic behavior of summation arithmetic functions. A probabilistic approach based on the use of the axioms of probability theory is used for these purposes. Sufficient conditions are proved under which these functions have a limiting normal distribution. Arithmetic functions having a limiting normal distribution are found in the paper. The author investigates summation functions of a general form and finds sufficient conditions under which they have a limiting normal distribution. Examples of arithmetic functions that satisfy these requirements are considered. We prove an ergodic theorem for summation arithmetic functions, for which the sequence of random variables has the stationarity property in the broad sense. The asymptotics of the growth of the deviation of the values of the arithmetic function from its mean value are investigated. It is shown that an equivalent formulation of the Riemann hypothesis for the Mertens function is satisfied almost everywhere.

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arXiv2017

Converting of algebraic Diophantine equations to a diagonal form with the help of an integer non-orthogonal transformation, maintaining the asymptotic behavior of the number of its integer solutions

The author showed that any homogeneous algebraic Diophantine equation of the second order can be converted to a diagonal form using an integer non-orthogonal transformation maintaining asymptotic behavior of the number of its integer solutions. In this paper, we consider the transformation to the diagonal form of a wider class of algebraic second-order Diophantine equations, and also we consider the conditions for converting higher order algebraic Diophantine equations to this form. The author found an asymptotic estimate for the number of integer solutions of the diagonal Thue equation of odd degree with an amount of variables greater than two, and also he got and asymptotic estimates of the number of integer solutions of other types of diagonal algebraic Diophantine equations.

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arXiv2017

Converting of algebraic Diophantine equations to a diagonal form with the help of generalized integer orthogonal transformation, maintaining the asymptotic behavior of the number of its integer solutions

The paper presents a new generalized integer orthogonal transformation which consists of a well known orthogonal transform followed by stretching the basis vectors maintaining the asymptotic behavior of the number of integer solutions for algebraic Diophantine equation. The author shows the properties of this transformation and he receives the algorithm for finding the matrix elements of a generalized integer orthogonal transformation for algebraic Diophantine equation of the second order to diagonal form. The article includes examples illustrating the reduction of algebraic equations of the second order to the diagonal form with the help of integer generalized orthogonal transformation and of determination asymptotics behavior of integer solutions for these equations.

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arXiv2017

Integer conversions and estimation of the number of integer solutions of algebraic Diophantine equations

The paper assesses the top number of integer solutions for algebraic Diophantine Thue diagonal equation of the degree $n \geq 2$ and number of variables $k > 2$ and equations with explicit variable in the case when the coefficients of the equation are of the opposite signs. The author found integer conversions that maintain the asymptotic behavior of the number of integer solutions of algebraic Diophantine equation in the case of the conversion equation to diagonal form. The paper considers the estimation of the number of integer solutions for some types of algebraic Diophantine equations with nondiagonal form.

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arXiv2016

Estimating of the number of integer (natural) solutions of inhomogeneous algebraic Diophantine diagonal equations with integer coefficients

This paper investigates the upper bound of the number of integer (natural) solutions of inhomogeneous algebraic Diophantine diagonal equations with integer coefficients without a free member via the circle method of Hardy and Littlewood. Author found the upper bound of the number of natural solutions of inhomogeneous algebraic Diophantine diagonal equations with explicit variable. He developed a method in the paper, which allows you to perform the low estimate of the number of natural (integer) solutions of algebraic Diophantine equation with integer coefficients. Author obtained a lower estimate (with this method) of the number of integer (natural) solutions for certain kinds of inhomogeneous algebraic Diophantine diagonal equations with integer coefficients with any number of variables (including Thue equation).

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arXiv2016

Estimating of the number of natural solutions of homogeneous algebraic Diophantine diagonal equations with integer coefficients

Author developed a method in the paper, which, unlike the circle method of Hardy and Littlewood (CM), allows you to perform a lower estimate for the number of natural (integer) solutions of algebraic Diophantine equation with integer coefficients. It was found the lower estimate of the number of natural solutions to various types of homogeneous algebraic Diophantine equations with integer coefficients diagonal form with any number of variables using this method. Author obtained upper bound of the number of the natural solutions (using CM) of one type of homogeneous Diophantine equation for values $k \geq \log_2 s$, where $k$ is the degree of the equation and $s$ is the number of variables. It was also found the upper bound of the number of the natural solutions of the homogeneous algebraic Diophantine equation with integer coefficients with a small number of variables. Author investigated the relations of upper and lower estimates of the number of natural solutions of homogeneous Diophantine equation with integer coefficients diagonal form in the paper.

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arXiv2015

Estimations of the number of solutions of algebraic Diophantine equations with natural coefficients using the circle method of Hardy-Littlewood

This article discusses the question - how to estimate the number of solutions of algebraic Diophantine equations with natural coefficients using Circular method developed by Hardy and Littlewood. This paper considers the estimate of the number of solutions of algebraic Diophantine equation: $c_1x_1^{k_1}+c_2x_2^{k_2}+...+c_sx_s^{k_s}=n$. The author found the asymptotic estimate for the number of solutions of this equation as a function of the value $n$, if all coefficients and $n$ are natural. This article analyzes the results and shows that these estimates of the number of natural solutions of the equations have high accuracy.

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