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For each global field K, we shall construct and study two basic arithmetic functions, M (K) σ (z) and its Fourier dual ˜ M (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 .C onnections 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.

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