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

Among the equivalents of the Riemann Hypothesis (RH), a millennium unsolved problem ([7]), the Alcantara-Bode equivalent ([2]) is of a particular interest due to its formulation: the Riemann Hypothesis holds iff the null space of the integral operator on L2(0, 1) having the kernel function the fractional part of the ratio (y/x), contains only the element 0 ([2]), i.e. iff the operator is injective. This equivalent formulation allows us to use techniques outside of the number theory field in order to prove it and so, to show that RH holds. We introduced in this article a functional analysis - numerical method for investigating the injectivity of the linear bounded operators on separable Hilbert spaces, updating and extending the result from [1]. The method is built around the result obtained (Theorem 1) saying that if a linear bounded operator is strict positive on a dense set, then it is injective. When the operator - or its associated Hermitian replacing it if needed, is only positive definite on a dense set, additional operator properties should be considered. Dealing with this case, two are the directions we choose for obtaining the corresponding criteria, using the operator approximations over finite dimension subspaces in L2(0, 1) whose union is dense: · involving the operator finite rank approximations on subspaces or, · involving its adjoint restrictions on subspaces. (Injectivity Criteria [1]). Applying both versions of the method on the dense set of indicator interval functions, we proved the Alcantara-Bode equivalent is true so, that RH holds. This solution for RH is not one in pure math. field as seems to have been expected since 1859. However, it is in line with the Clay Math Inst. principle that has been expressed (citing [7]) by: "A proof that it is true for every interesting solution would shed light on many of the mysteries surrounding the distribution of prime numbers."