If, after 28 months of waiting and the third request, I within a few hours receive “Reviewer # 1: Sorry that this review took so long. However, this paper does not contain any results of value”, I know that the referee read only the author's name and, possibly, the title of the article.
If I get...
No, if someone makes incredible claims and the work does not seem to have any chance of backing up those claims, then the reviewers have no obligation to waste their time looking for errors that the author is not going to recognize as such anyway.
Working off of @TobiasKildetoft's comments, it sounds like your efforts were poorly presented. Given that you purport to solve some very difficult problems, and given that your work isn't sufficiently readable, the reviewers decided it wasn't worth the trouble to find the errors. For an article to pass peer review, it isn't just necessary that the reviewers are unable to find errors, but also that they are able to understand the solutions and verify their accuracy.
I'm talking about the fact that reviewers can understand the solutions and check their accuracy, they should try to read the work, but not make assumptions about what is written in the article
About the readability of the work - it is available to the student of the school
Readability isn't about who is allowed to read the work, it's about whether it is written well enough that the people who read it can understand it. Understanding is not just something readers do, it is something that writers facilitate.
I spent several years writing this work just so that the reader can read it quickly and without difficulty. If the burden of writing lies with the author, this does not mean that the reader is freed from the burden of understanding. It is obvious that the reader must also make some effort
The problem is that no one has an obligation to read your manuscript. Referees are almost universally working for free, as a community service. Since experts should not value their time too low, there is always a question about whether or not to invest time in a given project/issue/enterprise. If, on general principles, a given task appears to be a dubious investment of time, one doesn't engage with it. Examination of attempted solutions of long-outstanding problems by people with no other track record are not obviously a good investment of time.
... and about "burden of understanding": of course, one should not expect to exert no effort. However, there should be limits to the types and quantities of effort required. For example, highly idiosyncratic or ambiguous notation or terminology (when there is already adequate terminology) is in itself a bad signal. When I am referee-ing (or grading Written Prelims for grad students), I exercise the principle that it should not be more difficult for me to understand what is written than it would be to write a solution/proof myself on blank pages...
... and I suspect that the reference to "feelings" was an attempt to be more polite, and not judgemental, not an emotional reaction.
I looked briefly at your paper arxiv.org/pdf/1402.6571.pdf. Indeed I do not find it easy to read, but since you asked for a particular problem point: your equation (15) gives an asymptotic for the number of twin primes up to n of the form C(n) n/(log n)^2. Except possibly for the value of C(n), this is the Hardy-Littlewood conjecture, so -- much stronger than the existence of infinitely many twin primes. But all you say in the way of justification is "Construction for double sieving can be described roughly as..." This is a (big) problem point in your paper.
I do not see a problem here. Equation (15) is given only to show how to go from $ \ pi_2 (n) $ to $ \ pi_g (n) $, where $ g = 2 ^ k $, and then to $ \ pi_g (n) $ for any even $ g $. Could you clarify what the problem is?
The problem is, as I said, that you assert something that implies infinitely many twin primes with no justification whatsoever. Then in the proof of Proposition 10 you write that pi_2(6m) >mH_m holds for all m> 5 without any justification. I showed your paper just now to a colleague of mine who is an expert in this area of number theory. He agreed that these are serious gaps in the argument. More explanation than this is not reasonable to ask: the burden has to be on you to make sense rather than on the mathematical community to explain to your satisfaction why you are not making sense.
@ Bryan Krause In the proof of Proposition 10, we define the expression H_m. This is important. Then, by numerical verification, we find that pi_2 (6m)> mH_m holds for all m> 5. In the same way, we determine the values of m in formulas (16) and (17), which are some analogues of the inequality obtained from the Rosser and Schoenfeld inequalities. I think that the values of x in the Rosser and Schoenfeld inequalities are obtained in the same way, but no one requires their justification. Or I did not understand the question?
@AndreiAllakhverdov You aren't pinging correctly, and you aren't even pinging the right person.
This type of math is far far far from my area of expertise but it seems like everything you are talking back falls back to (15) which you only justify by saying "Construction for double sieving can be described roughly as" - you give no other justification for that conclusion and do not explain the justification. Maybe the justification is in the previous paragraph? But you seem to only show it is true for a couple bounded examples. That's far from saying it is true for all n. I might be way off because, again, I know nothing of this sort of math.
@Andrei: "Then, by numerical verification, we find that pi_2 (6m)> mH_m holds for all m> 5." This includes the statement that you showed that there are infinitely many twin primes by "numerical verification." This is not possible. "I think that the values of x in the Rosser and Schoenfeld inequalities are obtained in the same way, but no one requires their justification." Rosser and Schoenfeld are giving "effective forms" -- i.e., with explicit constants -- of results that were already known asymptotically. This is not the same as what you are claiming at all...
...In short: you have major misunderstandings of the work of Rosser and Schoenfeld and also of what constitutes a valid mathematical proof. Because of this you have written paper "does not contain any results of value." So Reviewer #1 was right and your implication that he did not read your paper is not justified. You would be using your time much more efficiently if you shored up your knowledge of basic mathematics rather than submitting papers that claim to prove Goldbach, prime k-tuples, etc. You may not be able to understand why your work is wrong until you learn more.
By the way, this is really the last I will say on the matter, but I hope you will remember that professional mathematicians have engaged with and critiqued your work. "Really all mathematicians not only do not see, but also do not want to see beyond their nose?" No, not really.
@BryanKrause Thanks to you both. I understood you both. I guess I wrote the work poorly. Maybe I can change the style and write more transparently. Perhaps I worked too long on the article and many things that seem obvious to me are not obvious to the reader. In fact, the main idea of the work is to construct a double (and multiple) sieve of the Eratosthenes type. The Goldbach-Euler conjecture and the twin primes conjecture simply follow from this construction.
As for the basics of mathematics, you are wrong - I own them completely. I have a mathematical education, although I do not work in my specialty and can only study mathematics in my spare time. But I do not speak English at all.
Discussion on question by Andrei Alla…
Discussion on question by Andrei Alla…