On the Method for Proving RH Using the Alcantara-Bode Equivalence

Our result is in line with the Beurling and Alcantara-Bode equivalent formulations of the Riemann Hypothesis (RH). Also, it intends as a numerical method to supply the lack of the methods in literature for investigation of the injectivity of linear bounded operators on separable Hilbert spaces. The criteria exploit the operator approximation positivity properties on finite dimension subspaces having their union a dense set covering a wide range of linear, bounded operators. For operators that are not positive definite, taking their associated Hermitian, it consists of: a Hermitian Hilbert-Schmidt operator whose family of finite rank approximations built on a dense set having the positivity parameters inferior bounded, has a null space containing only 0, i.e. containing no not null elements. We obtained the injectivity for the Alcantara-Bode integral operator connected to Riemann Zeta function, that is in fact the equivalent formulation of the RH.