Are my calculations of a new constant similar to Mill's constant based on $\lfloor A^{2^{n}}\rfloor$ and Bertrand's postulate correct?
Solution 1:
It has been a long time since I wrote the above question, and I was able to gather a confirmation that the contents and calculations are correct. During the last year I tried to publish the contents as a very simple paper ($5$ pages long) in several (up to $7$ different) known journals (each one of them took from $1$ month to $4$ months to receive feedback since the paper was sent). All of them confirmed (via peer review) that the calculations are correct, but basically the general comment is that is a result that can be expected to happen, so it is not so interesting to be published.
I have learnt from the experience how to write correct papers (in American English) and how difficult is the professional world of publications. So let me say that now I can understand better the struggles of professional mathematicians to be able to publish new and interesting content (I admire you!).
I would like to share all the anonymous feedback I have received from the different peers that reviewed the paper in order to close the question. The more feedback I received the better the paper was getting in terms of completion. So for your viewing pleasure (some of them were extremely kind and useful to update the paper) and not in the same order that the paper was sent to the journals:
The author has a simpler theoretical foundation at the cost of less cute result. It is not a surprise that such thing can be done, and the proof is indeed very short and elementary.
You might consider submitting a version of your paper to Mathematics Magazine for their problems section.
You do not adequately survey the literature. There are many papers dealing with refinements of Mills' paper that you do not cite; for example, Matomäki, K., Prime-representing functions. Acta Math. Hungar. 128 (2010), no. 4, 307–314 already obtains a sequence with growth rate like $L^{2^n}$. And other papers, such as Baker, Roger C.The intersection of Piatetski-Shapiro sequences. Mathematika 60 (2014), no. 2, 347–362. do even better. (I reviewed and added the references, this was a very important update).
In principle the proof idea should work. All in all I do not find the paper illuminating enough for the () experts won't find the result surprising and non-experts won't get a clear idea of the topic from this manuscript.
The subject itself will be of interest to very few people - the Mills' constant itself is a function that uses the primes to construct primes, and thus does not really tell any useful or interesting things about primes (the same thing can be done with any sequence, not just the primes).
This is a promising project, but I am not convinced that it is successful. To be more precise, the construction in the present form is not complete. (Followed by some specific points that still were not clear in the paper and I was not able to fix properly at that moment).
Plus the classical answers:
I am sorry to say we will be unable to consider your manuscript for publication, because we currently have an excessively large backlog of articles awaiting publication. As a result we have been forced to drastically curtail submissions for the time being.
Your submission is unacceptable since the topics are too specialized.
So in general, it has been a good experience, I have found a lot of patient and kind peers, professionals that had a serious look to the paper (even if I was not a professional) and provided very useful insights and advises.
Finally I wanted to thank @Konstantinos Gaitanas the kind help and review during these months by email. Unfortunately I was not able to give a successful result to that effort in this occasion. But I hope that this experience will help other people to try again until success. Lessons learned.