A generalization of a conjecture of Hardy and Littlewood to algebraic number fields (Q5928669)

Define, for an even number \(d \in {\mathbb{N}}\), NEWLINE\[NEWLINEP_d(x):= |\{ 0 < n < x ;\;n, n+d \text{ both prime}\}|. NEWLINE\]NEWLINE In 1922 Hardy and Littlewood conjectured that NEWLINE\[NEWLINE \lim_{x \to \infty} \frac{P_d(x)}{P_2(x)}= \prod_{\substack{ p|d\\ p>2}} \frac{p-1}{p-2} NEWLINE\]NEWLINE and that \(P_2(x)\) is asymptotic to NEWLINE\[NEWLINE 2\prod_{p>2} \left( 1 - \frac{1}{(p-1)^2} \right) \int_2^x \frac{dy}{\log^2 y}. NEWLINE\]NEWLINE In the present paper the authors formulate these conjectures in the language of an algebraic number field. Moreover, they give some numerical computations for various types of number fields (imaginary quadratic, cubic, \dots) which confirm the plausibility of the stated formulae.