On the Method for Proving RH Using the Alcantara-Bode Equivalence (II)

This study was inspired by Alcantara-Bode’s equivalent to the Riemann Hypothesis published in 1993, the equivalent formulation consisting in the injectivity of an integral operator connected to Riemann Zeta function. Surprisingly, the research on this line has not continued, an explanation would be the lack of criteria for the injectivity of integral operators. This paper aims to fill this gap by proposing a functional-numerical analysis solution exploiting the operator positivity properties on dense sets. The main theorem says that a linear, bounded operator strict positive definite on a dense set of a separable Hilbert space, has its null space containing only the null element, equivalently, it is injective. Having in mind to obtain a generic and useful criterion, we gradually changed the hypothesis of the strict positivity of the operator on a dense set to the involvement at the end, of the associated Hermitian operator that is semi positive on the whole space requesting additional properties related to the positivity of operator approximations on finite dimension subspaces. Then, in order to apply the criterion for Hermitian Hilbert-Schmidt operators, we choose an adequate dense set allowing to obtain operator sparse matrix representations. The criterion applied to the associated Hermitian of the Alcantara-Bode integral operator, showed that the equivalent holds, so the Riemann Hypothesis is true.