Virtual characters on L-functions

Abstract

In this expository note we show the inception and development of the Heilbronn characters and their application to the holomorphy of quotients of Artin L-functions. Further we use arithmetic Heilbronn characters introduced by Wong, to deal with holomorphy of quotients of certain L-functions, e.g,, L-functions associated to CM elliptic curves. Furthermore we use the supercharacter theory introduced by Diaconis and Isaacs to study Artin L-functions associated to such characters. We conclude the note surveying about various other unconditional approaches taken based on character theory of finite groups.