# On the question about a strengthening of Bertrand's postulate

I am trying to work out what I think about the suitability of this question. The author is in a very real and awkward situation: His$^{\ast}$ graduate student has a result. The result is not that exciting, but it is not clearly superseded by prior results. (I'll back these statements up at the end of this question.)

The student has done the right thing: Asked his advisor whether the result is worthwhile. Unfortunately, the advisor also doesn't know the conventions of the field well enough to answer. One answer is that the advisor should e-mail friends of his who are closer to the relevant field, but this can take a long time without a good answer, even if he has such friends. He can simply encourage the student to submit the result and see what happens.

The question for us is, do we want "post the question on MO" to be a third option? The advantage I see is that we have readers who are very familiar with the publication standards of number theory journals. I think the argument against is that the question is seen as too subjective, and is usually the sort of thing that appears on academia.SE (where the question started) not here.

I've cast a vote to reopen, but I'd be glad to hear arguments both ways.

$^{\ast}$ Gender presumed on the basis of the poster's choice of pseudonym.

As i707107 computes, the result follows from the prime number theorem with error term $x/(\log x)^3$. But the question says that the proof only uses Rosser's theorem which looks to me to roughly be equivalent to a one-sided estimate $\pi(x) < (1+\epsilon) x/\log x$. There is a standard trick to convert one-sided estimates into two-sided estimates, but it seems surprising to me that the proof needs two powers of $\log x$ less than expected.

Is that surprise worth publishing? I don't know. How much more elementary is Rosser than getting $x/(\log x)^N$ error? How interested are number theory journals in new proofs of Bertrand like results which follow immediately from PNT with strong bounds? I feel like this is the sort of question that only people familiar with the publishing practices of analytic number theory journals can answer.

• The question feels made up to me. Let us not waste time on this. At least let us wait until OP shows up on this site. – user9072 May 17 '14 at 21:23
• It is not clear to me that truth is being told here. Among other things, the fact that student is a graduate in one sentence and an undergraduate in another makes me wonder. I will stop short of calling the poster a liar and assume a mistake is made, but if I wanted to craft something just to raise interest level on MathOverflow, and did not care about representing myself honestly, I might start with a story resembling that in the post. I recommend more caution and skepticism than usual for this post. Gerhard "Experiencing A Credibility Gap Here" Paseman, 2014.05.17 – Gerhard Paseman May 17 '14 at 21:37
• The choice of names (Hilbert, Gauss), reminds me of another user that at some point went by Cantor. – Andrés E. Caicedo May 17 '14 at 22:18
• Let me just briefly comment on a the matter in general: Regardinless the special context here, I think it is a bad idea to allow this type of question. We were always rather careful not to allow people asking "is my result good" and "is my result correct" type of questions. But this would be exactly this. There (hypothetically) being some senior mathematician as a filter in between might help a bit, but only that much. And also it will be impossible to tell and enforcen, and would open the floodgates to every amateur asking about their latest break through. – user9072 May 17 '14 at 23:26
• Even locally the story is not really plausible. For example, on the one hand we should believe OP is not familiar with the matter on the other hand they just say "Rosser's theorem" assuming eveybody knows what is meant. This does not fit. Then not locally, there is the strange coincidence of somebody asking about this result just days ago (with a in some sense similar username). From what was asked there it does not even seem related. But then should it really be by chance? Finally, look at the account's questions on academia.SE. (cont.) – user9072 May 18 '14 at 0:04
• They have a nephew an undergraduate that wants to publish something in number theory and need to ask something basic about publishing. Yet they claim to have graduate students. But also on academia.SE they need an endorser for arXiv. But then they are sned many manuscript by amateurs. So we have some quite clueless yet still somehow senior mathematician. All this seems quite unlikely. But in any case to give a maningfull answer to the question asked a lot more details on the math are needed. If they should provide them we can recondiser until then it is a NARQ (even believing everything). – user9072 May 18 '14 at 0:08
• The post itself is riddled with markers that make it suspect to me. On the mathematics, why is n>4 used as a qualifier? Unless they have a different version of c(n), 1<2<4=c(1), 2<3<6=c(2), 3<5<8, 4<7<9, 5<7<10, 6<11<12, and so on. The second statement was noted to be a restatement of pi(c(n)-1) = c(n) - n -1, the "significant" result. The real meat is the suggestion of an elementary proof that c(n) grows fast enough to have a prime between n and c(n). As quid suggests, more detail is needed to answer the question honestly. Gerhard "Even If It Isn't Honest" Paseman, 2014.05.17 – Gerhard Paseman May 18 '14 at 0:28
• I took a break from MO because I thought the moderators were being too soft on this kind of crap. I poke my head back in and here we are. Sigh. – Felipe Voloch May 18 '14 at 0:37
• @quid's comment pointed out the user's account on academia.SE: here's a link academia.stackexchange.com/users/14733/alfred-gauss which should be compared with math.stackexchange.com/users/135953/william-hilbert – Yemon Choi May 18 '14 at 0:50
• @FelipeVoloch Generally I feel, and I think some of the other moderators do too, that it's the community who ought to be allowed to decide what's crap and what merits consideration. (Speaking even as one of the more "activist" moderators in terms of deciding whether or not to close/delete questions.) Sorry that we can't please everyone. FWIW, I would have voted to close myself, if I had ordinary 10k+ privileges and not the awesome Thor-like moderation powers I now find myself possessing. :-) – Todd Trimble May 18 '14 at 1:16
• @FelipeVoloch Tell me about it! There are a great many off-topic questions and instances of bad behavior that I and my colleagues adjudicate every day (with great reliance on the watchful eyes of the community), or otherwise tricky situations to deal with. It's a fairly big job and we do have lives outside MO. But yeah, getting off-topic here. :-) – Todd Trimble May 18 '14 at 1:55
• Incidentally, in academia.stackexchange.com/revisions/20674/1 Alfred Gauss said he had received a paper from an Indian undergraduate named Sayantan Roy, who berated Gauss for behaving like the mathematicians who ignored Ramanujan's letters. Google's search index includes the page mathoverflow.net/users/48849/sayantan-roy, which now leads to mathoverflow.net/users/48849/william-hilbert, and Bing's index also thinks William Hilbert used to call himself Sayantan Roy. – Henry Cohn May 18 '14 at 3:24
• What a soap-opera we have going here! – Ryan Budney May 18 '14 at 4:15
• Further to my previous comment, compare academia.stackexchange.com/questions/20674 and math.stackexchange.com/questions/754430/… and matheducators.stackexchange.com/questions/1983 (Apologies to David Speyer if he'd rather this thread be left to rest) – Yemon Choi May 19 '14 at 18:52
• @YemonChoi No problem. I was not aware of the multiple identities and strange history here. The closure jumped out at me because I can well imagine myself posting a question along the lines of "my student has found the following variant on the Dold-Kan theorem (say). I am not very familiar with this field. Is this already in the literature and/or already folklore?" – David E Speyer May 19 '14 at 19:28