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On $\log L$ and $L'/L$ for $L$-functions and the associated "$M$-functions": Connections in optimal cases
Yasutaka Iharaa,
Kohji Matsumotob a RIMS, Kyoto University, Kyoto, Japan
b Graduate School of Mathematics, Nagoya University, Nagoya, Japan
Abstract:
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).
Key words and phrases:
$L$-function, value distribution, mean value theorem, arithmetic Dirichlet series, function field over finite field.
MSC: Primary
11R42; Secondary
11M38,
11M41 Received: October 20, 2009
Language: English
DOI:
10.17323/1609-4514-2011-11-1-73-111