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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(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 ζ K (σ +τ i), where σ is fixed and τ varies, will also be briefly discussed.

We study the value-distribution of Dirichlet L-functions L(s, χ) in the half-plane σ = s > 1/2. The main result is that a certain average of the logarithm of L(s, χ) with respect to χ, or of the Riemann zeta-function ζ(s) with respect to s, can be expressed as an integral involving a density function, which depends only on σ and can be explicitly constructed. Several mean-value estimates on L-functions are essentially used in the proof in the case 1/2 < σ ≤ 1.

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