Bounded gaps between product of two primes in imaginary quadratic number fields

TL;DR

This paper investigates the distribution of gaps between products of two primes in imaginary quadratic number fields, establishing the existence of infinitely many close pairs under certain class number conditions using advanced sieve methods.

Contribution

It introduces a novel approach combining GGPY and Maynard methods to study prime products in quadratic fields, proving the existence of infinitely many close pairs.

Findings

Existence of infinitely many pairs of prime products with small differences in quadratic fields.

Bounded differences between prime product pairs depend on the class number of the field.

Method extends sieve techniques to algebraic number fields for prime product gaps.

Abstract

We study the gaps between products of two primes in imaginary quadratic number fields using a combination of the methods of Goldston-Graham-Pintz-Yildirim \cite{GGPY}, and Maynard \cite{MAY}. An important consequence of our main theorem is existence of infinitely many pairs $\alpha_1, \alpha_2$ which are product of two primes in the imaginary quadratic field $K$ such that $|\sigma(\alpha_1-\alpha_2)|\leq 2$ for all embedding $\sigma$ of $K$ if the class number of $K$ is one and $|\sigma(\alpha_1-\alpha_2)|\leq 8$ for all embedding $\sigma$ of $K$ if the class number of $K$ is two.