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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 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 thedoi:10.2977/prims/1216238306 fatcat:elo6ef2pzrd3velz6bs7akqj2i