Joint Universality

In this chapter, we shall prove a conditional joint universality theorem for functions in S. Joint universality means that we are concerned with simultaneous uniform approximation, a topic invented by Voronin [362, 364]. Of course, such a result cannot hold for an arbitrary family of L-functions: e.g., ζ(s) and ζ(s)² cannot be jointly universal. The L-functions need to be sufficiently independent to possess this joint universality property. We formulate sufficient conditions for a family of L-functions in order to be jointly universal and give examples when these conditions are fulfilled; for instance, Dirichlet L-functions to pairwise non-equivalent characters (this is an old result of Voronin) or twists of L-functions in the Selberg class subject to some condition on uniform distribution.