An introduction to the Bernoulli function

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 $\operatorname{B}(s, v) = - s\, ζ(1-s, v)$ 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 $ζ$ and $ξ$ 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 André numbers. The Euler function is introduced as the difference between values of the Hurwitz-Bernoulli function. The André function and the Seki function are the unsigned versions of the extended Euler resp. Bernoulli function.