An introduction to the Bernoulli function

TL;DR

This paper introduces and studies a generalized Bernoulli function linked to the zeta function, exploring its properties, functional equations, and connections to classical number sequences like Bernoulli, Euler, and André numbers.

Contribution

It presents a novel extended Bernoulli function based on integral representations, connecting classical results with modern zeta function analysis.

Findings

Defines the Bernoulli function independently of the zeta function.

Establishes the functional equation and representations involving Riemann zeta and xi functions.

Recovers classical results and extends Bernoulli numbers to odd indices and rational numbers.

Abstract

We explore a variant of the zeta function interpolating the Bernoulli numbers based on an integral representation suggested by J. Jensen. The Bernoulli function $\operatorname{B}(s, v) = - s\, \zeta(1-s, v)$ can be introduced independently of the zeta function if it is based on a formula first given by Jensen in 1895. We examine the functional equation of $\operatorname{B}(s, v)$ and its representation by the Riemann $\zeta$ and $\xi$ function, and recast classical results of Hadamard, Worpitzky, and Hasse in terms of $\operatorname{B}(s, v).$ The extended Bernoulli function defines the Bernoulli numbers for odd indices basing them on rational numbers studied by Euler in 1735 that underlie the Euler and Andr\'{e} numbers. The Euler function is introduced as the difference between values of the Hurwitz-Bernoulli function. The Andr\'{e} function and the Seki function are the unsigned versions of the extended Euler resp. Bernoulli function.