An introduction to the Bernoulli function

arXiv2009.06743MaRDI QIDQ6349086

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Publication date: 11 September 2020

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 $operatornameB(s,v)=-s,zeta(1-s,v)$ 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 $operatornameB(s,v)$ and its representation by the Riemann $zeta$ and $xi$ function, and recast classical results of Hadamard, Worpitzky, and Hasse in terms of $operatornameB(s,v).$ 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.

Has companion code repository: https://github.com/PeterLuschny/BernoulliFunction

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