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

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.