On $\log L$ and $L'/L$ for $L$-functions and the associated "$M$-functions": Connections in optimal cases

Let $\mathcal L(s,\chi)$ be either $\log L(s,\chi)$ or $L'/L(s,\chi)$, associated with an (abelian) $L$-function $L(s,\chi)$ of a global field $K$. For any quasi-character $\psi\colon\mathbb C\to\mathbb C^\times$ of the additive group of complex numbers, consider the average "$\mathrm{Avg}_{\mathfrak f_\chi=\mathfrak f}$" of $\psi(\mathcal L(s,\chi))$ over all Dirichlet characters $\chi$ on $K$ with a given prime conductor $\mathfrak f$. This paper contains (i) study of the limit as $N(\mathfrak f)\to\infty$ of this average, (ii) basic studies of the analytic function $\tilde M_s(z_1,z_2)$ in 3 complex variables arising from (i) (here, $(z_1,z_2)\in\mathbb C^2$ is the natural parameter for $\psi$), and (iii) application to value-distribution theory for $\{\mathcal L(s,\chi)\}_\chi$. Our base field $K$ is either a function field over a finite field, or a special type of number field: the rational number field $\mathbb Q$ or an imaginary quadratic field. But in the number field case, the Generalized Riemann Hypothesis is assumed in (i) and (iii).