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.

arXiv admin note: substantial text overlap with arXiv:1803.03218

References in corpus (1)

• An explicit bound for the least prime ideal in the Chebotarev density theorem