On the Injectivity of an Integral Operator Connected to Riemann Hypothesis

Abstract This version is a straightforward and more intuitive approach, pointing out the logic behind the method ([5],[6]) used to investigate the injectivity of linear operators on separable Hilbert spaces. On a dense family of including finite dimension subspaces, we highlighted the connection between the projections of the elements outside the dense family and the condition numbers of the linear operator restrictions to the family subspaces. The connection we established is: for a zero of unitary norm of an operator linear strict positive definite on the dense family, on each subspace the ratio between its norms of the reqsiduum and the corresponding projection is majorizing the condition number of the operator restriction. So, we proved that a criteria for injectivity is the existence of a strict positive inferior bound for the sequence of the condition numbers associated to the operator restrictions on the subspaces. To address the Alcantara-Bode equivalent formulation ([1]) of the Riemann Hypothesis using the criteria, we choose a suitable dense family of subspaces on the separable Hilbert space L$^2(0,1)$ well known in literature built on indicator interval functions with disjoint support. Numerically, we obtained a 'multigrid' structure of the finite dimension including approximation subspaces that together with the orthogonal projection schema defined in [3] are fitting the requests of the method introduced. As result we obtained ([5], [6]): the integral operator used by Alcantara-Bode in its equivalent formulation of RH ([1]) has the condition numbers sequence bounded by a constant, showing that the operator used in this equivalence is injective. Then we could claim: the Riemann Hypothesis holds provided that its equivalent formulation is true.