On “$M$-Functions” Closely Related to the Distribution of $L^{\prime }/L$-Values

Abstract

For each global field $K$, we shall construct and study two basic arithmetic functions, $M_{\sigma }^{(K)}(z)$ and its Fourier dual ${\tilde{M}}_{\sigma }^{(K)}(z)$, on $C$ parametrized by $\sigma >1/2$. These functions are closely related to the density measure for the distribution of values on $C$ of the logarithmic derivatives of $L$-functions $L(\chi ,s)$, where $s$ is fixed, with $\text{Re}(s)=\sigma$, and $\chi$ 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 ${\zeta }_K(\sigma +\tau i)$, where $\sigma$ is fixed and $\tau$ varies, will also be briefly discussed.