  Rademacher symbols may be defined in terms of Dedekind sums, and give the
value at zero of the zeta function associated to a narrow ideal class of a real
quadratic field. Duke extended these symbols to give the zeta function values
at all negative integers. Here we prove Duke's conjecture that these higher
Rademacher symbols are integer valued, making the above zeta value denominators
as simple as the corresponding Riemann zeta value denominators. The proof uses
detailed properties of Bernoulli numbers, including a generalization of the
Kummer congruences.
