Computing the residue of the Dedekind zeta function

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 $8.33\log \Delta _K/(\sqrt {X}\log X)$, where $\Delta _K$ 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.