  The aim of this paper is to give a full detail of the proof given by Harder
of a theorem on the denominator of the Eisenstein class for
$\mathrm{SL}_2(\mathbb{Z})$ and to show that the theorem has some interesting
applications including the proof of a recent conjecture by Duke on the
integrality of the higher Rademacher symbols. We also present a sharp universal
upper bound for the denominators of the values of partial zeta functions
associated with narrow ideal classes of real quadratic fields in terms of the
denominator of the values of the Riemann zeta function.
