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On L-functions over function fields: Power-means of error-terms and distribution of L'/L-values

(1.1.3) Avgχ (mod f)|gPy(s, χ)| tends to 0 as y 7→ ∞ uniformly with respect to integral ideals f and to s ∈ C such that Re(s) ≥ 1/2 + e (Theorem A, §2.2). Here, χ runs over the (suitably normalized) non-principal characters mod f . The proof is based on the ideas and techniques used in [3] applied to the situation of the function field case. As an application (of the case of f ′ P (s, χ)), we shall give a sharpened version of Theorem 7 of [1], to the effect that the function Mσ(z) constructed there is, in fact, the density function for the distribution of values of {L′(s, χ)/L(s, χ)}χ in a strong sense. Here, s ∈ C is fixed with σ = Re(s), and χ runs over a suitably normalized family of Dirichlet characters on K with prime conductors. The only conditions for σ is, now, σ > 1/2