In the framework of Distributions and Pseudo Differential Operator theory the Hilbert-transformed Theta-function is used to verify the Hilbert-Polya conjecture in a weak form, i.e. the non-imaginary solutions E(n) of Zeta(1/2+iE(n))=0 are the eigenvalues of an appropriate Hermitian operator H.
All attempts failed so far to represent Riemann’s duality equation as transform of a self-adjoint integral operator due to the (too strong) analytical regularity of the underlying Theta function.
The basic idea of the proof is applying the concepts of Hyperfunctions and Hilbert transforms applied to the Theta function to get a weak, self-adjoint formulation of the Riemann duality equation:
Riemann´s duality equation is equivalent to the Jacobi-Theta-function property, which is the Poisson summation formula for the Gauss-Weierstrass density function f(x). The regularity of Jacobi´s Theta function is given by the regularity of f(x), i.e. it is analytical, which isn´t the case for f(1/x). This leads to divergent Mellin transform integrals of a corespondingly built self adjoint integral operator in the critical stripe. Riemann`s proof of the duality equation basically leads to a replacement of the pure f structure to a form xdf/dx to manage the convergence of the Mellin integrals properly. The prize being paid for this are all the failures so far mentioned above.
The key idea of this proof is to replace the Poisson summation formula G(x) for f(x) by the Hilbert transform of it instead, which ensures convergent corresponding Mellin integrals of H(G(x)) and H(G(1/x))/x at least in a weak form, when applying Müntz formula to build Zeta(s) and Zeta(1-s) in the critical stripe. The prize to be paid to ensure integral convergences is an only weak representation of Riemann´s duality equation in the calculus of variations, enabled by complex-valued Pseudo Differential Operator theory and the hyper function. Nevertheless, spectral theory can be applied to give the positive answer to the Berry conjecture for the critical line, by which the validity of Riemann´s hypothesis follows.
Here we are (last update 09.12.2010):
There are only formal changes to the following original version from February 2010:
Dualization of Poisson Summation formula
For a first flavor about duality in the context of the Poisson summation formula we refer to