On the least prime ideal and Siegel zeros

TL;DR

This paper establishes an effective bound on the smallest prime ideal in a ray class group of a number field, especially when a Siegel zero exists, with applications to primes represented by quadratic forms.

Contribution

It provides a new effective Linnik-type bound for prime ideals in ray class groups considering Siegel zeros, with explicit constants and uniformity in the number field.

Findings

Effective bounds for least prime ideals in ray class groups.

Explicit constants in the bounds considering Siegel zeros.

Application to primes represented by binary quadratic forms.

Abstract

Let $K$ be a number field, $\mathfrak{q}$ be an integral ideal, and $\mathrm{Cl}(\mathfrak{q})$ be the associated ray class group. Suppose $\mathrm{Cl}(\mathfrak{q})$ possesses a real exceptional character $\psi$, possibly principal, with a Siegel zero $\beta$. For $\mathcal{C} \in \mathrm{Cl}(\mathfrak{q})$ satisfying $\psi(\mathcal{C}) = 1$, we establish an effective $K$-uniform Linnik-type bound with explicit constants for the least norm of a prime ideal $\mathfrak{p} \in \mathcal{C}$. A special case of this result is related to rational primes represented by certain binary quadratic forms.