  The main purpose of this paper is to show some relations between the Riemann
zeta function and the generalized Bernoulli polynomials of level $m$. Our
approach is based on the use of Fourier expansions for the periodic generalized
Bernoulli functions of level $m$, as well as quadrature formulae of
Euler-Maclaurin type. Some illustrative examples involving such relations are
also given.
