The least prime ideal in a given ideal class

Abstract

Let $K$ be a number field with the discriminant $D_K$ and the class number $h_{K}$, which has bounded degree over $\mathbb{Q}$. By assuming GRH, we prove that every ideal class of $K$ contains a prime ideal with norm less than $h_{K}^2\log(D_K)^{2}$ and also all but $o(h_K)$ of them have a prime ideal with norm less than $h_{K}\log(D_K)^{2+\epsilon}$. For imaginary quadratic fields $K=\mathbb{Q}(\sqrt{D})$, by assuming Conjecture~\ref{piarcor} (a weak version of the pair correlation conjecure), we improve our bounds by removing a factor of $\log(D)$ from our bounds and show that these bounds are optimal.