UPPER BOUNDS OF ROOT DISCRIMINANT LOWER BOUNDS

For any rational number t is an element of [0, 1], define the logarithmic Martinet function beta(t) to be the liminf of the logarithm of the root discriminant of number fields K with r(1) (K) = [K : Q] = t as [K : Q] goes to infinity. Under the generalized Riemann hypothesis for Dedekind zeta functions of number fields, we show that beta(t) < 14.55 for a dense subset of rational numbers t is an element of [0, 1]. We also study unconditional estimates of the growth of root discriminants by studying how the polynomial discriminant behaves under perturbation of coefficients, and by using Pisot numbers.