Lower Bounds for Regulators of Number Fields in terms of their Discriminants

We prove inequalities that compare the regulator of a number field with its absolute discriminant. We refine some ideas in Silverman's work in 1984 where such general inequalities are first proven. In order to prove our main theorems, we combine these refinements with the authors' recent results on bounding the product of heights of relative units in a number field extension.
Comment: A version of this article has been accepted for publication in Journal de Th\'eorie des Nombres de Bordeaux. arXiv admin note: substantial text overlap with arXiv:2008.06124