# On “<em>M</em>-Functions” Closely Related to the Distribution of <em>L'</em>/<em>L</em>-Values

### Yasutaka Ihara

Kyoto University, Japan

## Abstract

For each global ﬁeld *K*, we shall construct and study two basic arithmetic functions, *Mσ*(*K*)(*z*) and its Fourier dual *M~σ*(*K*)(*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 *ζK*(*σ* + *τi*), where *σ* is ﬁxed and *τ* varies, will also be brieﬂy discussed.

## Cite this article

Yasutaka Ihara, On “<em>M</em>-Functions” Closely Related to the Distribution of <em>L'</em>/<em>L</em>-Values. Publ. Res. Inst. Math. Sci. 44 (2008), no. 3, pp. 893–954

DOI 10.2977/PRIMS/1216238306