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 $$\alpha _1, \alpha _2$$ α 1 , α 2 which are product of two primes in the imaginary quadratic field K such that $$|\sigma (\alpha _1-\alpha _2)|\le 2$$ | σ ( α 1 - α 2 ) | ≤ 2 for all embeddings $$\sigma $$ σ of K if the class number of K is one and $$|\sigma (\alpha _1-\alpha _2)|\le 8$$ | σ ( α 1 - α 2 ) | ≤ 8 for all embeddings $$\sigma $$ σ of K if the class number of K is two.