Dedekind sums and class numbers of imaginary abelian number fields

Abstract

As a consequence of their work, Bruce C. Berndt and Ronald J. Evans in 1977 and Larry Joel Goldstein and Michael Razar in 1976 obtained a formula for the square of the class number of an imaginary quadratic number field in terms of Dedekind sums. Here we focus only on this formula and give a new short and simple proof of it. We finally express the relative class numbers of imaginary abelian number fields in terms of Dedekind sums.