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 (Proc Lond Math Soc 98:741–774, 2009), and Maynard (Ann Math 181:383–413, 2015). 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)|le 2 for all embeddings sigma of K if the class number of K is one and |sigma (alpha _1-alpha _2)|le 8 for all embeddings sigma of K if the class number of K is two.