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

TL;DR

This paper investigates the distribution of prime numbers, analyzes the error in probabilistic assumptions about prime density, and develops methods for asymptotic analysis of arithmetic functions in probabilistic number theory.

Contribution

It introduces new forms of the law of large numbers for arithmetic functions and provides a method to find asymptotics of their probabilistic characteristics.

Findings

Error estimates for prime density assumptions

New analogues of the law of large numbers for arithmetic functions

Method for asymptotic analysis of probabilistic characteristics

Abstract

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.