The Values of the Zeta-Function of a Class of Ideals and the Ankeny, Artin and Chowla Congruences for the Class Number of Real Quadratic Number-Fields

AbstractLet K = Q(√D) be a real quadratic number field with discriminant D > 0, χ = χD the Dirichlet character belonging to K (χ(n) = (D/n)) and p an arbitrary prime number ≥ 5. The natural numbers T and U are defined by the power ϵ = ϵ1 − χ(p)p0 = 12(T + U√D) of the fundamental unit ϵ0 > 1 of K. We obtain the congruence 1p(p − χ(p)) NK/Q(ϵ)TUhD ≡ 2Bp − 1,χ mod p between the class number hD of K and the (p − 1)st generalized Bernoulli number belonging to χ. In the case p | D we have ϵ = ϵ0 and the congruence (1) can be written in the form NK/Q(ϵ0) TUhD ≡ 2Bp − 1,χ mod p. Using the Kummer congruences for the generalized Bernoulli numbers it is easy to show that (2) is essentially the well known congruence of Ankeny, Artin, and Chowla for the class number of a real quadratic number field. Similar congruences are true for the prime number p = 3