Representing ideal classes of ray class groups by product of prime ideals of small size

Abstract

We prove that, for every modulus $\mathfrak{q}$, every class of the narrow ray class group $H_{\mathfrak{q}}(\mathbf{K})$ of an arbitrary number field $\mathbf{K}$ contains a product of three unramified prime ideals $\mathfrak{p}$ of degree one with $\mathfrak{Np}\le (t(\mathbf{K})\mathfrak{Nq}{)}^3$, where $t(\mathbf{K})$ is an explicit function of $\mathbf{K}$ described in the paper. To achieve this result, we first obtain a sharp explicit Brun-Titchmarsh Theorem for ray classes and then an equally explicit improved Brun-Titchmarsh Theorem for large subgroups of narrow ray class groups. En route, we deduce an explicit upper bound for the least prime ideal in a quadratic subgroup of a narrow ray class group and also for the size of the least ideal that is a product of degree one primes in any given class of $H_{\mathfrak{q}}(\mathbf{K})$.