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The main purpose of this paper is to show some relations between the Riemannzeta function and the generalized Bernoulli polynomials of level $m$. Ourapproach is based on the use of Fourier expansions for the periodic generalizedBernoulli functions of level $m$, as well as quadrature formulae ofEuler-Maclaurin type. Some illustrative examples involving such relations arealso given.

Some relations between the Riemann zeta function and the generalized Bernoulli polynomials of level $m$