Some days ago I have asked this question but I got no response. Will it be ok if I post this question in this site (I am asking this because crossposting isn't generally welcomed in this site)? Also can some suggestion be given so that I may edit the question to make it more suitable for this site?


It is now answered.

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    $\begingroup$ The question looks generally alright (to me) for MO, but can I ask how you know such a value $f(\epsilon)$ exists for all $\epsilon > 0$? For all I know this is well-known to experts, but a link would be very nice. $\endgroup$
    – Todd Trimble Mod
    Nov 21 '14 at 13:00
  • $\begingroup$ @ToddTrimble: See en.wikipedia.org/wiki/Legendre%27s_constant . $\endgroup$ Nov 21 '14 at 13:46
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    $\begingroup$ I don’t understand the point of the question. On the one hand, it asks for one function with particular properties (which shouldn’t be hard to find in the relevant literature, I expect), on the other hand, it insists on a “general method”. $\endgroup$ Nov 21 '14 at 13:53
  • $\begingroup$ @EmilJeřábek Yes, now I feel very silly for asking. :-( And now on second thought, I'm with Emil and not quite sure what the question is asking, since for this particular question the extraction of $f$ should be routine. Maybe someone should answer at MSE... $\endgroup$
    – Todd Trimble Mod
    Nov 21 '14 at 14:08
  • $\begingroup$ @ToddTrimble: Can you elaborate where particularly is your difficulty in understanding the question? $\endgroup$
    – user57432
    Nov 21 '14 at 14:10
  • $\begingroup$ @EmilJeřábek: Can you give some link of the literature dealing with this problem? $\endgroup$
    – user57432
    Nov 21 '14 at 14:14
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    $\begingroup$ Let's put it this way: would you be satisfied if someone wrote down an explicit function $f$ such that $\frac{x}{\log(x) - (1-\epsilon)} < \pi(x) < \frac{x}{\log(x) - (1+\epsilon)}$ for all $x > f(\epsilon)$ (with some indication of proof)? My difficulty is knowing if "general method" is supposed to extend beyond this particular problem (if so, you would need to elaborate the class of problems, and this is highly unclear at the moment). $\endgroup$
    – Todd Trimble Mod
    Nov 21 '14 at 14:20
  • $\begingroup$ @user170039: For example, see the very Dusart’s thesis that you linked to in the question, in particular Théorème 1.10, points 6,7. With a bit of elementary algebra, this will give you a bound $\epsilon\le c/\log x$ for some explicit constant $c$, wherefrom you can take $f(\epsilon)=\exp(c/\epsilon)$. (This is optimal bound up to the value of $c$.) $\endgroup$ Nov 21 '14 at 14:36
  • $\begingroup$ @ToddTrimble: Yes. You are right. My main objective was to write down an explicit function $f$. $\endgroup$
    – user57432
    Nov 21 '14 at 16:42
  • $\begingroup$ I have now responded at MSE with an answer. Emil was spot on. $\endgroup$
    – Todd Trimble Mod
    Nov 21 '14 at 20:25
  • $\begingroup$ Well, so did I. $\endgroup$ Nov 21 '14 at 20:27
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    $\begingroup$ @nicael, please define "usually". (for me, m.se always stands for Mathematics Stackexchange) $\endgroup$ Nov 27 '14 at 5:03
  • $\begingroup$ @GerryMyerson okay, but how is this relevant; m.se is not identical to MSE. :-) [this is not a pure joke, as I actually would associate m.se more with Math and MSE with Meta] $\endgroup$
    – user9072
    Nov 27 '14 at 12:05
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    $\begingroup$ @quid, since I don't associate anything with Meta Stack Exchange, I associate all variants of mse with mathstackexchange. $\endgroup$ Nov 27 '14 at 12:17
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    $\begingroup$ @GerryMyerson but maybe you should, and adopt a more efficient naming scheme. Anyway, I am pretty sure that the predominant usage (throughout the network) is that MSE stands for Meta Stack Exchange, which you can take as the definition of "usually" that you asked for. $\endgroup$
    – user9072
    Nov 27 '14 at 23:56

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