Joint Universality

• Published in: Value-Distribution of L-Functions pp. 229 - 248
• Format: Book Chapter
• Language: English
• Published: Berlin, Heidelberg Springer Berlin Heidelberg
• Series: Lecture Notes in Mathematics
• ISBN: 3540265260; 9783540265269
• ISSN: 0075-8434; 1617-9692
• DOI: 10.1007/978-3-540-44822-8_12

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.