For each global field K, we shall construct and study two basic arithmetic functions, M (K) σ (z) and its Fourier dualM (K) 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

more »

... logarithms or (not much studied) logarithmic derivatives of ζ K (σ +τ i), where σ is fixed and τ varies, will also be briefly discussed.