On log L and L'/L for L-Functions and the Associated “M-Functions”: Connections in Optimal Cases

Abstract

Let L(s; ) be either logL(s; ) or L 0 =L(s; ), associated with an (abelian) L-function L(s; ) of a global eld K. For any quasi- character : C ! C of the additive group of complex numbers, con- sider the average \Avgf =f of (L(s; )) over all Dirichlet characters on K with a given prime conductor f. This paper contains (i) study of the limit as N(f) ! 1 of this average, (ii) basic studies of the ana- lytic function ~ Ms(z1; z2) in 3 complex variables arising from (i) (here, (z1; z2) 2 C 2 is the natural parameter for ), and (iii) application to value-distribution theory for fL(s; )g . Our base eld K is either a function eld over a nite