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 Original Research Article

Let K = image(√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 image of the fundamental unit ε0 > 1 of K. We obtain the congruence image 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 image 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.