Probabilistic analytical approach to determining the asymptotics of prime objects on the initial interval of the natural series

TL;DR

This paper introduces a new probabilistic-analytical method for deriving asymptotics of prime objects within initial natural numbers, providing insights consistent with existing prime number conjectures.

Contribution

A novel probabilistic-analytical approach based on constructing a probability space to analyze prime objects and their asymptotics, aligning with known conjectures.

Findings

Derived asymptotics for twin primes and Goldbach pairs

Method yields estimates matching prime number conjectures

Provides an alternative probabilistic framework for prime conjecture analysis

Abstract

This paper considers a probabilistic-analytical approach to determining asymptotics of prime objects on the initial interval of the natural series. The author proposes a new method based on the construction of a probability space. An arithmetic function is considered that counts the number of prime objects (for example, prime numbers, twin primes, prime values of polynomials) on a given probability space. The properties of this arithmetic function are proved. Based on these properties, the specified probabilistic-analytical approach is constructed. As examples of the application of this approach, the definition of the asymptotics of the number of prime twins and pairs of prime numbers that add up to an even number (based on Goldbach's conjecture) is considered. It is shown that the proposed method allows one to obtain asymptotic estimates that coincide with the known conjecturea of prime number theory. This approach opens up new possibilities for studying conjectures about prime numbers, offering an alternative way to prove them based on probabilistic methods.