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

The Alcantara-Bode equivalent (1993) obtained from the Beurling equivalent formulation (1955) of the Riemann Hypothesis (RH), a Millennium Problem, states that RH holds if and only if the null space of a specific integral operator on $L^2(0,1)$ related to the Riemann Zeta function, contains only the null element or equivalently, the operator is injective. Their equivalent formulations allow solutions outside the area of the number theory. In order to prove the Alcantara-Bode equivalent, we presented a method for investigating the injectivity of linearly bounded operators built on a generic result introduced (Theorem 1): a linear bounded operator on a separable Hilbert space strict positive definite on a dense set is injective. Then, we introduced its versions on finite dimension approximation subspaces whose union is dense, updating and extending the results from [1] by separating the analysis of the operator restrictions from their operator approximations on finite-dimension subspaces. The positivity of such operator approximations on a family of subspaces, ensures the strict positivity of the operator on the dense set provided that the sequence of the positivity parameters is inferior bounded. The criteria introduced in [1] reformulated in the new context is useful when no information we have to consider operator approximations. We proved the Alcantara-Bode equivalent applying this method, having as effect the solution of RH that is, the Riemann Hypothesis holds.