Siegel zeros and their consequences in number theory

Abstract

We shall discuss two consequences of the existence of Siegel zeros to arithmetic. The first is regarding class numbers of quadratic fields and here the existence of Siegel zeros is an impediment to proving optimal results. On the other hand, the second is a rather surprising theorem of Heath-Brown which says that the existence of Siegel zeros implies the twin prime conjecture. We shall be brief with proofs and often skip the gory technical lemmas. If we have time, we shall discuss the Deuring-Heilbronn zero repulsion phenomenon.