Note on the Contribution of Low Zeros to Weil's Explicit Formula for Minimal Discriminants

Abstract

AbstractThe bulk of this paper consists of tables giving lower bounds for discriminants of number fields up to 48. The lower bounds are obtained by using two different inequalities for the discriminant, one due to Odlyzko, and the other due to Serre. These inequalities are derived from Weil's explicit formula by choosing suitable weight functions. The bounds are compared with actual values of the discriminants, and the relative errors are computed. The computations show that, at least for values computed, the bounds obtained via Odlyzko's inequality are better than those obtained via Serre's inequality, and are generally within a few percentage points of the true value. This difference can be attributed to a difference in the weighting given to the contribution of low zeros by the two inequalities.