On logL and L′/L for L-functions and the associated “M-functions”: Connections in optimal cases

Let L(s, χ) be either logL(s, χ) or L′/L(s, χ), associated with an (abelian) L-function L(s, χ) of a global field K. For any quasicharacter ψ : C → C× of the additive group of complex numbers, consider the average “Avgfχ=f” of ψ(L(s, χ)) over all Dirichlet characters χ on K with a given prime conductor f. This paper contains (i) study of the limit as N(f) → ∞ of this average, (ii) basic studies of the analytic function M̃s(z1, z2) in 3 complex variables arising from (i) (here, (z1, z2) ∈ C is the natural parameter for ψ), and (iii) application to value-distribution theory for {L(s, χ)}χ. Our base field K is either a function field over a finite field, or a special type of number field: the rational number field Q or an imaginary quadratic field. But in the number field case, the Generalized Riemann Hypothesis is assumed in (i) and (iii). 2000 Math. Subj. Class. Primary: 11R42; Secondary: 11M38, 11M41.