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 $G_K(X) = \max \{r > 0: \text{there exists } x_0 \in O_K \text{ such that } |N_K(x_0)| \leq X \text{ and } B(x_0,r) \text{ contains no primes} \}$. We show that $ G_{K}(X) \gg_K (\log X) \frac{\log_2(X) \log_4(X)}{\log_3(X)}$.