2. Other considerations and reader comments

Taking a look at the formula (5), we observe that: the left side is the injectivity parameter, an intrinsic operator restriction property, and that the right side is an evaluation formula of the approximation of the residuum of an eligible element, this evaluation being independent of any operator restriction.

This observation suggests that once we know that the linear operator verifies (5) for any eligible element (being injective), then any linear operator verifying for every n, is injective too. This was discussed in one of our preprints.

There exist injective linear operators having the injectivity parameters sequence unbounded inferior – such an example is provided in our paper. This means that the zeros of a linear operator strict positive definite on a dense family are among those eligible elements which verify (4), however, it is not certain if all eligible elements verifying (4) are zeros of the operator. Unfortunately, our injectivity criteria does not cover such cases.

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Remarks

1. The solution we provided for solving Alcantara-Bode’s equivalent formulation of the Riemann Hypothesis is based on appl. math. methods. This may not be fully agreed with by some pure math specialists, who might expect an elegant solution in the pure math. field. Nevertheless, we consider that the Riemann Hypothesis  has been solved.
2. For unknown reasons, the Abstract and the Full-Text versions of my article, published in JPAM ([3], DOI: 10.37532/2752-8081.22.6(4).19-23), are still not fully corrected, even if I keep asking the publisher to implement the required fixes.
3. Due to the fact that JPAM journal ceased to make corrections to my article in Oct. ’22, I cannot ask for the following errata: “The family of functions , k = 1,n, nh =1, defines a trace class integral operator with the kernel function” should be replaced with “The family of functions , k = 1,n, nh =1, defines a finite rank integral operator with the kernel function”, (par. 3. ‘Approximations of the Integral Operators’). Note that par. 5, (‘Some numerical aspects of the method’,) uses the correct formulation. I apologize.
4. The DOI (JPAM) is not always working. As a result, the cross reference of the published article and the preprint is not working properly. For this reason, even if the article has been published almost one year ago, Research Square considers that “it has not been peer reviewed by a journal”. However, despite Research Square’s statement that my article has not been peer reviewed, the JPAM web page specifies: Received: 12-Jul-2022, Manuscript No. puljpam-22-5145; Editor assigned: 13-Jul-2022, Pre QC No. puljpam-22-5145 (PQ);  Reviewed: 22-Jul-2022 QC No. puljpam-22-5145 (Q); Revised: 25-Jul-2022, Manuscript No. puljpam-22-5145 (R); Accepted Date: Jul 28, 2022; Published: 30-Jul-2022, DOI: 10.37532/2752-8081.22.6(4).19-23

Note: I wish to express my gratitude to Terence Tao’s blog followers and to the editors of the IMA Journal for their spontaneous reactions.