The value-distribution of artin L-functions associated with cubic fields in conductor aspect

Arising from the factorizations of the Dedekind zeta-functions of cubic fields, we obtain Artin L-functions of certain two-dimensional representations. In this paper, we study the value-distribution of such Artin L-functions for families of non-Galois cubic fields in conductor aspect. For this, we apply asymptotic formulas for the counting functions of cubic fields proved by Bhargava et al. (Invent Math 193 (2):439-499, 2013) or Taniguchi and Thorne (Duke Math J 162(13): 2451-2508, 2013). Then the Artin L-functions associated with cubic fields are connected with random variables called the random Euler products. We construct a density function for the random Euler product, whose Fourier-Laplace transform has an infinite product representation. Furthermore, we prove that various mean values of the Artin L-functions are represented by integrals involving this density function. By the class number formula, the result is applied to the study on the distribution of class numbers of cubic fields. We show a formula for the sum involving the class numbers which is regarded as a cubic analogue of Gauss' formula on quadratic class numbers.