On the Method for Proving the RH Using the Alcantara-Bode Equivalence
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Among the equivalents of the Riemann Hypothesis (RH), a millennium unsolved problem ([10]), one is of a particular interest due to its surprising and almost elementary expression in terms of the functional analysis: Riemann Hypothesis holds if and only if 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. if and only if the operator is injective. As we do not find in the literature references about how to approach a problem of this type, we introduced here a solution containing the theory and related methods regarding the injectivity of linear, bounded operators on separable Hilbert spaces. The generic theorem introduced saying that an linear bounded operator strict positive definite on a dense set is injective is the basis on which we built our solution. Next, we showed that an Hilbert-Schmidt operator with the finite rank operator approximations on a family of finite dimension subspaces is njective if its positivity parameters on the family subspaces are inferior bounded. It could work also for the operator restrictions on the subspaces if the operator is only positive definite provided that we involve the adjoint operator. The injectivity of the Hilbert-Schmidt integral operator that is the counterpart in the Alcantara-Bode equivalence ([2]) of RH, has been proved taking the dense set of indicator interval functions in L2 (0, 1). Consequently, from the Alcantara-Bode equivalence, RH is true: the Riemann Zeta function defined by the infinite sum ζ(s) = ∑∞ n=11/n s has its non trivial zeros on the vertical line σ = 1/2.