On “M-functions” closely related to the distribution of L’/L-values

[Abstract] For each global field K, we shall construct and study a basic arithmetic function M (K) σ (z) on C parametrized by σ > 1/2, together with its Fourier transform M̃ (K) σ (z). This function M (K) σ (z) is 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 characters on K.