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 The result is not that exciting, but it is not clearly superseded by prior results.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.