On certain mean values of logarithmic derivatives of $L$-functions and the related density functions

Abstract

We study some "density function" related to the value-distribution of L-functions. The first example of such a density function was given by Bohr and Jessen in 1930s for the Riemann zeta-function. In this paper, we construct the density function in a wide class of L-functions. We prove that certain mean values of L-functions in the class are represented as integrals involving the related density functions.

On the value-distribution of the difference between logarithms of two symmetric power L-functions

“…Inspired by the idea of Guo, Mine [29] proved the existence (and the explicit construction) of the M -function for (ζ ′ K /ζ K )(s) in t-aspect, where ζ K (s) denotes the Dedekind zeta-function of an algebraic number field K (including the non-Galois case), with an explicit error estimate in the limit formula of the form (3.1). In [31], he extended the result to the case of more general L-functions, belonging to a certain subclass of M .…”