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

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.