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Victor Volfson

Publications and source records attributed to Victor Volfson.

At least 19 recordsLinked to original sources
arXiv2026

Estimating the tail of the singular product for the Hardy Littlewood and Bateman Horn conjectures

This paper investigates the asymptotic behavior of the tail of the singular product arising in the Hardy Littlewood and Bateman Horn conjectures for one dimensional systems of polynomials. A universal estimate is proved, showing that the contribution of large primes decays like the reciprocal of the logarithm, regardless of the structure of the system. For linear systems (trivial Galois group) superfast convergence is obtained. For nonlinear systems a coefficient is defined that is expressed via the average over the Galois group; in the abelian case and under the Riemann Hypothesis for Dirichlet L functions a more precise error estimate is obtained. Mixed systems containing both linear and nonlinear polynomials are also considered. Numerical experiments, presented as summary tables, confirm the theoretical conclusions. The results provide a rigorous theoretical foundation for computing singular series and refine the Bateman Horn formula.

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arXiv2026

Effective Bounds for Singular Series in the Multivariate Bateman Horn Conjecture

We propose an approach to estimating the error in computing the singular series in the multivariate Bateman Horn conjecture, based on a combination of methods from algebraic geometry and analytic number theory. For general polynomial systems, we establish a uniform estimate in primes for the local factors, from which we derive a universal upper bound for the relative error expressed in terms of a geometric constant depending on the Betti numbers of the projective closures of the hypersurfaces. For a single polynomial, an explicit bound for this constant is given in terms of the degree and the number of variables, making the result constructive. In the diagonal case, using Katz exact formula for diagonal cohomologies, we obtain substantially faster convergence; an additional application of the Hardy Littlewood circle method allows us to further refine the estimate. Numerical examples show that diagonal systems yield an accuracy gain of several orders of magnitude compared with the general case. Our results provide rigorous quantitative error control and demonstrate that the convergence rate is determined not only by the degree but also by the geometric structure of the polynomial system.

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arXiv2026

Generalization of the Hardy-Littlewood conjecture to almost-prime number tuples

The article presents a generalization of the classical Hardy-Littlewood conjecture concerning the density of prime tuples to the case of tuples consisting of almost-prime numbers (numbers with a specified quantity of prime divisors). The work investigates tuples of natural numbers where each element is subject to an individual factorization requirement. A proposed asymptotic formula for the quantity of such tuples is presented, where the density is determined by the product of two constants: the standard Selberg constant, which depends solely on the tuple pattern, and a correction factor, which depends only on the set of requirements for the number of prime divisors at each position in the tuple. The author proves that the admissibility of a pattern for prime numbers implies its admissibility for almost-prime numbers. The principle of symmetry is established - the correction factor depends only on the multiset of requirements, not on the order of elements within the tuple. An empirical-analytical method for calculating the correction factor is developed, based on the invariance of the Selberg constant under pattern stretching. The method is tested on tuples of small length (pairs and triples), for which tables of calculated coefficients with high accuracy are provided. The method is justified in the work.

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arXiv2026

Dependencies of prime numbers in a tuple

This paper investigates the dependence between primes in tuples through the analysis of the Hardy-Littlewood constant. A detailed analysis of the behavior of the constant for the pattern $(0,d)$ is conducted, depending on the arithmetic properties of $d$, including cases of convergence to the twin prime constant, divergence to infinity, and oscillatory behavior. Using methods from analytic number theory, the limiting distributions of the corresponding multiplicative functions are studied. It is shown that for symmetric tuples, the Hardy-Littlewood constant monotonically decreases as the tuple length decreases, indicating a weakening dependence between the primes. Theoretical conclusions are supported by calculations for specific symmetric tuples.

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arXiv2024

Some questions of connection between summation functions and the corresponding Dirichlet series

The paper proves a generalization of Wintner's theorem on the asymptotics of summation functions to the case of summation functions with nonlinear asymptotics. The class of arithmetic functions that have a logarithmic asymptotic mean is studied. The Kronecker lemma is generalized to the case when the corresponding Dirichlet series diverges. Several assertions are proven and illustrative examples are given.

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arXiv2023

An approach to determining the existence of a limit distribution of additive arithmetic functions

An approach will be proposed to determine the existence of a limit distribution of additive arithmetic functions in this work. It is based on assertions that will be proven in this work and on the properties of Dirichlet convolution and M\"obius inversion. Based on this approach, formulas will be obtained for finding the mean and variance for the limit distribution of additive arithmetic functions. Examples of this approach to proving the existence of a limit distribution of additive arithmetic functions and finding the mean value and variance for the limit distribution of additive arithmetic functions are considered.

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arXiv2023

Summation functions with nonlinear asymptotic behavior

The paper considers a universal approach that allows one to quite simply obtain nonlinear asymptotic estimates of various summation functions. It is shown the application of this approach to the asymptotic estimation of divergent Dirichlet series. Several assertions have been proven and numerous examples have been considered.

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arXiv2022

Maximum distance between consecutive primes and other related questions

The paper considers the asymptotic of the ratio of the number of primes not exceeding the primorial and the number of residues in the reduced system of residues for the given primorial. We study the relationship between asymptotic lower bounds for the values of the Jacobstal function and the maximum distance between successive primes. One algorithm for computing the Jacobstal function is given. The substantiation of the conjectures about the upper estimate of the maximum distance between successive prime numbers is given.

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arXiv2022

Estimation of the asymptotic behavior of summation functions

The paper considers asymptotics of summation functions of additive and multiplicative arithmetic functions. We also study asymptotics of summation functions of natural and prime arguments. Several assertions on this subject are proved and examples are considered.

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arXiv2022

Asymptotic behavior "almost everywhere" of additive and multiplicative arithmetic functions

We define the asymptotic behavior "almost everywhere" of additive and multiplicative arithmetic functions in the paper. Classes of additive and multiplicative arithmetic functions are singled out for which the asymptotics coincides "almost everywhere" with the asymptotics of the corresponding strongly additive and multiplicative arithmetic functions. Several assertions are proved and examples are considered.

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arXiv2021

Asymptotics of sums of functions of primes located on an arithmetic progression

We investigate the problem of the distribution of sums of functions of prime numbers located on an arithmetic progression. This problem is closely related to the problem of the distribution of prime numbers on an arithmetic progression. Based on this distribution, a general formula was obtained for the asymptotic estimate of the sums of functions of primes, and also asymptotics was found for the sums of various functions of primes on a geometric progression. Several assertions about asymptotics of sums of functions of prime numbers on a geometric progression are proved. Necessary and sufficient conditions for the existence of these asymptotics are also proved.

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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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arXiv2021

An estimate of asymptotics of the moments of additive arithmetic functions with a limit distribution defined on a subset of the natural series

We study the asymptotics of the moments of arithmetic functions that have a limit distribution, not necessarily normal, defined on a subset of the natural series that satisfies certain requirements. Several assertions are proved on estimating the asymptotics of the moments of strongly additive arithmetic functions and also with additive functions of the class H that have a limit distribution and are defined on a subset of the natural series. The first version of the article is devoted to the study of the asymptotics of the moments of arithmetic functions that have a limit distribution on the natural series. The second version of the article is devoted to the study of the asymptotics of the moments of arithmetic functions that have a limit distribution on an arithmetic progression.

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arXiv2021

Additive arithmetic functions with limit normal distribution

This paper proves several assertions on sufficient conditions for the convergence of additive arithmetic functions to the normal distribution. A generalization of the Erdos-Kac theorem was proved and determines the rate of convergence of additive arithmetic functions to the normal distribution in the cases considered. The paper provides examples of using the proved assertions.

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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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