arXiv subjects

Bertrand Barrau

Publications and source records attributed to Bertrand Barrau.

2 recordsLinked to original sources

arXiv2013

Hilbert-Polya conjecture and Generalized Riemann Hypothesis

Extending a classical integral representation of Dirichlet L-functions associated to a non trivial primitive character we define associated functions B(y,z) which are eigenfunction of a Hermitian operator H. The eigenvalues are the imaginary parts of the L-functions zeros. We prove that if s is a non trivial zero of such a Dirichlet L-function with Re(s)<1/2, then: - the associated eigenfunction B(z,y) is square integrable. - the operator H is "Hermitian" for this function: = . We deduce from this (using the idea of Hilbert-Polya and finding a contradiction) the Generalized Riemann Hypothesis: the non trivial zeros of a Dirichlet L-function lie on the critical line Re(s)=1/2. This results correspond to a weak form of the Hilbert-Polya conjecture (as for Re(s)=1/2 the eigenfunctions presented here are not square integrable).

math.GM↗

arXiv2013

On Hilbert-Polya conjecture: Hermitian operator naturally associated to L-functions

Using as starting point a classical integral representation of a L-function we define a familly of two variables extended functions which are eigenfunctions of a Hermitian operator (having imaginary part of zeros as eigenvalues). This Hermitian operator can take also other forms, more symetric. In the case of particular L-functions, like Zeta function or Dirichlet L-functions, the eigenfunctions defined for this operator have symmetry properties. Moreover, for s zero fo Zeta function (or Dirichlet L-function), the associated eigenfunction has a specific property (a part of eigenfunction is cancelled). Finding such an eigenfunction, square integrable due to this "cancellation effect", would lead to Riemann Hypothesis using Hilbert-Polya idea.

math.NT↗