A Conjecture which Implies the Riemann Hypothesis

Abstract

In previous work [5] an interpretation of the Euler product for Dirichlet zeta functions was given in the theory of certain Hilbert spaces whose elements are entire functions. A matrix generalization of the Euler product results. The matrix product has similar convergence problems as the scalar product in the region where information about zeros of zeta functions is wanted. A new construction of Hilbert spaces of entire functions is made in the present work. The resulting matrix product has symmetry properties about a horizontal line passing through a given zero of the zeta function. A conjecture is made about the behavior of the matrix product on the line which has consequences for the behavior of the scalar product on the line. The Euler product for the zeta function [formula] reads ζ χ(s) −1 = ∏ (1 − x(p)p −s). The conjecture implies that the identity |ζ χ(s) −1| = ∏ |1−x(p)p −s| is valid when s lies on the horizontal half-line to the right of a zero in the critical strip. The conjecture implies the Riemann hypothesis for all Dirichlet zeta functions.