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

TL;DR

This paper extends the Hardy-Littlewood conjecture to almost-prime number tuples, providing an asymptotic formula, proving pattern admissibility transfer, and developing a method to calculate correction factors with empirical validation.

Contribution

It generalizes the classical conjecture to almost-primes, introduces a correction factor, and establishes a symmetry principle and calculation method for these factors.

Findings

Admissibility of prime patterns implies admissibility for almost-primes.

A symmetry principle states the correction factor depends only on the multiset of requirements.

A validated empirical-analytical method for calculating correction factors is presented.

Abstract

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.