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Volume 44, Issue 3, 2008, pp. 893–954
DOI: 10.2977/prims/1216238306

Published online: 2008-09-30

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

Yasutaka Ihara[1]

(1) Kyoto University, Japan

For each global field K, we shall construct and study two basic arithmetic functions, Mσ(K)(z) and its Fourier dual M~σ(K)(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 ζK(σ + τi), where σ is fixed and τ varies, will also be briefly discussed.

Keywords: L-functions, density function, Euler product, Bessel functions

Ihara Yasutaka: On “M-Functions” Closely Related to the Distribution of L'/L-Values. Publ. Res. Inst. Math. Sci. 44 (2008), 893-954. doi: 10.2977/prims/1216238306