Counting ideals in ray classes

TL;DR

This paper provides an explicit bound for the error term in counting ideals within ray class groups of number fields, refining Tatuzawa's 1973 asymptotic results with fully explicit constants.

Contribution

It establishes a fully explicit error bound for the asymptotic count of ideals in ray class groups, improving upon previous implicit bounds.

Findings

Explicit error bounds for ideal counts in ray class groups

Refinement of Tatuzawa's asymptotic with explicit constants

Enhanced understanding of distribution of ideals in number fields

Abstract

Let $\mathbf{K}$ be a number field and $\mathfrak{q}$ an integral ideal in $\mathcal{O}_{\mathbf{K}}$. A result of Tatuzawa from 1973, computes the asymptotic (with an error term) for the number of ideals with norm at most $x$ in a class of the narrow ray class group of $\mathbf{K}$ modulo $\mathfrak{q}$. This result bounds the error term with a constant whose dependence on $\mathfrak{q}$ is explicit but dependence on $\mathbf{K}$ is not explicit. The aim of this paper is to prove this asymptotic with a fully explicit bound for the error term.