Some Relations Between the Riemann Zeta Function and the Generalized Bernoulli Polynomials of Level $m$

Abstract

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.

Keywords

• bernoulli polynomials
• euler-maclaurin quadrature formulae
• generalized bernoulli polynomials of level $m$
• quadrature formula
• riemann zeta function