Computing the residue of the Dedekind zeta function

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 K by a clever use of the splitting of primes p > X p>X , with an error asymptotically bounded by 8.33 log ⁡ Δ K / ( X log ⁡ X ) 8.33\log \Delta _K/(\sqrt {X}\log X) , where Δ K \Delta _K is the absolute value of the discriminant of K 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 2.33 . This results in substantial speeding of one part of Buchmann’s class group algorithm.