Prime-free discs in imaginary quadratic fields

Abstract

Suppose $K$ is an imaginary quadratic field, and let $N_K$ denote the field norm in the ring of integers $O_K$. Let $B(x_0,r)=\{x\in O_K:|N_K(x-x_0)|<r\}$. Let $G_K(X)=\max \{r>0:\text{there exists }{x}_0\in O_K\text{ such that }|N_K(x_0)|\le X\text{ and }B(x_0,r)\text{ contains no primes}\}$. We show that $G_K(X){\gg }_K(\log X)\frac{{\log }_2(X){\log }_4(X)}{{\log }_3(X)}$.