Bernoulli functions, Hilbert-type poisson summation formulae, partial fraction expansions, and Hilbert-Eisenstein series

• Authors: Paul L. Butzer
• Venue: Analysis, combinatorics and computing
• Year: 2002

The aim of this article is to study the many properties of the Bernoulli functions and Bernoulli numbers, namely the Bα(z) and Bα(0) not for integral indices α = n but for complex alpha, introduced by the author and two collaborators in the past decade, and to place them in a broad setting. This includes the setting of generating functions, of fractional calculus, of replicative functions, of Dirichlet series and partial fraction expansions, of Eisenstein series, of Poisson, Boole, and Maclaurin summation formulae, together with their Hilbert -transform versions. The Bernoulli functions are connected with Stirling functions of the second kind, with two particular Dirichlet L-functions, and with the generalized Riemann (Hurwitz) zeta function; their functional equations are related to one another. A basic application of the classical Bα(0) is Euler's formula, a closed representation of the zeta function, ζ(s), at the even integers s = 2m, yielding that ζ(2m) is irrational. The evaluation of ζ(s) at odd integers s is a fundamental open problem. In fact it is only known that ζ(3) is irrational. It is shown that ζ(α) can be represented in terms of Bα∼(0) for α ∈ C, Re α 1, in particular ζ(2m + 1) in terms of the conjugate Bernoulli polynomial B2m+1∼(0) for odd integers s = 2m + 1. Here Bα∼(x) is the Hilbert transform of the (periodic) Bα(x). A good part of the article is devoted to a detailed examination of B2m+1∼(0), using the various summation formulae in order to obtain further information about ζ(2m + 1). The article contains several open problems in connection with Bernoulli functions and a new function, the Ω function, central to the field. Hopefully it may incite an interest in Hilbert-transform methods in analytic number theory, in particular in Hilbert-Eisenstein series.