Computing the residue of the Dedekind zeta function

TL;DR

This paper improves the method for computing the residue of the Dedekind zeta function of a number field under GRH, reducing error bounds and speeding up class group computations.

Contribution

It introduces a new approach using prime splitting and Weil's explicit formula to significantly improve error bounds in residue calculation under GRH.

Findings

Error bound improved from 8.33 to 2.33

Enhanced speed in Buchmann's class group algorithm

Utilizes Weil's explicit formula for better results

Abstract

Assuming the Generalized Riemann Hypothesis, Bach has shown that one can calculate the residue of the Dedekind zeta function of a number field K by a clever use of the splitting of primes p < X, with an error asymptotically bounded by 8.33 log D_K/(\sqrt{X}\log X), where D_K is the absolute value of the discriminant of K. Guided by Weil's explicit formula and still assuming GRH, we make a different use of the splitting of primes and thereby improve Bach's constant to 2.33. This results in substantial speeding of one part of Buchmann's class group algorithm.