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

This is the final version of the preprint series dedicated to provide a robust functional-numerical analysis method for investigation of the injectivity of linear bounded operators on separable Hilbert spaces. The aim has been to solve the Alcantara-Bode equivalent formulation of the Riemann Hypothesis. This equivalent 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. Or, such problems could be solved outside the area of the number theory. Our solution, a functional-numerical method is 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. We updated and extended the results from [1] by separating the analysis of the operator restrictions from their operator approximations on finite-dimension subspaces. With the numerical analysis criteria we reduced the injectivity problem to the existence of a strict positive bound modulo the square of the mesh of the positivity parameters defined as the diagonal entries in the sparse matrices associated to the integral operator. Applying this method we were able to prove the Alcantara-Bode equivalent having as effect the proof of RH that it holds.