# Publications of the Research Institute for Mathematical Sciences

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**Volume 44, Issue 3, 2008, pp. 893–954**

**DOI: 10.2977/prims/1216238306**

Published online: 2008-09-30

On “*M*-Functions” Closely Related to the Distribution of *L'*/*L*-Values

^{[1]}(1) Kyoto University, Japan

For each global ﬁeld *K*, we shall construct and study two basic arithmetic functions, *M _{σ}*

^{(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

*ζ*(

_{K}*σ*+

*τi*), where

*σ*is ﬁxed and

*τ*varies, will also be brieﬂy discussed.

*Keywords: **L*-functions, density function, Euler product, Bessel functions

Ihara Yasutaka: On “*M*-Functions” Closely Related to the Distribution of *L'*/*L*-Values. *Publ. Res. Inst. Math. Sci.* 44 (2008), 893-954. doi: 10.2977/prims/1216238306