  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.
