9 research outputs found
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).Comment: 16 Pages. Article withdraw as function Bs presented is not square
integrable as claimed. (Mistake is on one sub-domain considered: function I2
near y=0