Dependencies of prime numbers in a tuple

TL;DR

This paper explores the dependence between primes in tuples by analyzing the Hardy-Littlewood constant, revealing how the dependence varies with tuple structure and arithmetic properties, supported by theoretical and computational evidence.

Contribution

It provides a detailed analysis of prime tuple dependencies through the Hardy-Littlewood constant, including new insights into how these dependences change with tuple length and structure.

Findings

Dependence between primes varies with tuple length and structure.

The Hardy-Littlewood constant decreases as tuple length decreases.

Theoretical results are supported by calculations for specific symmetric tuples.

Abstract

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.