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

## TL;DR

This paper explores the connections between the Riemann zeta function and generalized Bernoulli polynomials of level m using Fourier analysis and Euler-Maclaurin quadrature, providing new relations and examples.

## Contribution

It introduces novel relations between the Riemann zeta function and generalized Bernoulli polynomials of level m through Fourier expansions and quadrature formulas.

## Key findings

- Derived new relations between zeta function and Bernoulli polynomials
- Presented Fourier expansion techniques for generalized Bernoulli functions
- Provided illustrative examples demonstrating these relations

## 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.