Ankeny–Artin–Chowla congruence
In number theory, the Ankeny–Artin–Chowla congruence is a result published in 1953 by N. C. Ankeny, Emil Artin and S. Chowla. It concerns the class number h of a real quadratic field of discriminant d > 0. If the fundamental unit of the field is
with integers t and u, it expresses in another form
for any prime number p > 2 that divides d. In case p > 3 it states that
where and is the Dirichlet character for the quadratic field. For p = 3 there is a factor multiplying the LHS. Here
represents the floor function of x.
A related result is that if d=p is congruent to one mod four, then
where Bn is the nth Bernoulli number.
There are some generalisations of these basic results, in the papers of the authors.
