The least prime ideal in a given ideal class

TL;DR

Under GRH and certain conjectures, the paper establishes bounds on the norms of the smallest prime ideals in each ideal class of a number field, with improvements for imaginary quadratic fields.

Contribution

The paper proves explicit bounds on the least prime ideal in each ideal class under GRH and conjectural assumptions, refining previous results and demonstrating optimality in specific cases.

Findings

Every ideal class contains a prime ideal with norm less than $h_K^2 ext{log}(D_K)^2$ under GRH.

All but a negligible fraction of classes have a prime ideal with norm less than $h_K ext{log}(D_K)^{2+ ext{epsilon}}$.

For imaginary quadratic fields, bounds are improved by removing a $ ext{log}(D)$ factor, and these bounds are shown to be optimal.

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.