Dependencies of prime numbers in a tuple

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.