The paper [10] contains a concise presentation of the proof of the Alcantara-Bode equivalence to RH. Citing the abstract:
“A linear bounded operator on a separable Hilbert space strict positive on a dense set is injective (Theorem 1, Par 2.). The result has been used as the backup for the criteria exploiting the operator approximation positivity properties on finite dimension subspaces having their union a dense set.
The functional-numerical methods introduced are a consequence of the observation that, when the dense set is an infinite union of finite dimension subspaces from a family, then the strict positivity on each subspace
of the operator approximations will attract the strict positivity on the dense set (Theorem 2) of the original operator provided that the positivity parameters of approximations are bounded by a strict positive constant.
The criteria applied to the Alcantara-Bode integral operator connected to Riemann Zeta function, showed that its null space contains only the null element. That is in fact the equivalent formulation of the Riemann Hypothesis.”
About RH - a Millennium Problem 