A mean value theorem for cubic fields

AbstractLet r(n) denote the number of integral ideals of norm n in a cubic extension K of the rationals, and define S(x)=∑n⩽xr(n) and Δ(x)=S(x)−αx where α is the residue of the Dedekind zeta function ζ(s,K) at 1. It is shown that the abscissa of convergence of∫0∞Δ(ey)2e−2yσdyis 13 as expected