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 $\operatorname{B}(s, v) = - s\, \zeta(1-s, v)$ 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.