there is a representation of the Zeta function as Mellin transform of the fractional part function. The idea of B.Nyman was about a proper translation of the RH into properties of this fractional part function. This criterion can be formulated in a purely Functional Analysis Hilbert space framework (B. Bagchi) using weighted l2-spaces defined by the fractional part function with argument (k/n).
The note below contains two parts:
- a Zeta fake representation as Mellin transform of the Hilbert-transformed fractional part function, which has all its zero on the critical line and is identical to the Zeta function in a weak sense. This gives a 2nd proof of the RH, following the same arguments as in the proof of the the Riemann Hypothesis in the related section above. The Nyman/Bagchi criterion is not applied.
- a proposal to move from a Fourier- to a Bessel-Fourier framework to build an appropriate dual space, which is most probably isomorph to the Bagchi's weighted l2-space. This Hilbert space is built by a stepwise density (bounded variation) function, defining a proper Stieltjes integral, which is isomorph to a Hilbert space with negative scale factor. The inverse formula from Stieltjes gives the corresponding hyper function representation of the corresponding Green function. The Lommel polynomials build the underlying orthogonal basis to that density function.
The rational to propose Lommel polynomials is, because its log(1/x)-singularity behavior is "closer" related to the Zeta function, than the "standard" Fourier and Hermite polynomials.
Here were are:
The note above was motivated by this paper
Main results concerning Lommel polynomials are recalled from the papers:
Papers from the references linking between Lommel-Bessel, Log-Gamma and Zeta functions are:
First published version: January 9, 2011