# On “$M$-Functions” Closely Related to the Distribution of $L_{′}/L$-Values

### Yasutaka Ihara

Kyoto University, Japan

## Abstract

For each global ﬁeld $K$, we shall construct and study two basic arithmetic functions, $M_{σ}(z)$ and its Fourier dual $M~_{σ}(z)$, on $C$ parametrized by $σ>1/2$. These functions are closely related to the density measure for the distribution of values on $C$ of the logarithmic derivatives of $L$-functions $L(χ,s)$, where $s$ is ﬁxed, with $Re(s)=σ$, and $χ$ runs over a natural inﬁnite family of Dirichlet or Hecke characters on $K$. Connections with the Bohr–Jessen type value-distribution theories for the logarithms or (not much studied) logarithmic derivatives of $ζ_{K}(σ+τi)$, where $σ$ is ﬁxed and $τ$ varies, will also be brieﬂy discussed.

## Cite this article

Yasutaka Ihara, On “$M$-Functions” Closely Related to the Distribution of $L_{′}/L$-Values. Publ. Res. Inst. Math. Sci. 44 (2008), no. 3, pp. 893–954

DOI 10.2977/PRIMS/1216238306