In another question on mathoverflow (What is the best currently proven bounds on prime gaps?) the following bound on the prime gap was quoted: $G(X)\ll \frac{X^{0.525}}{\log X}$ How do you prove this, and what paper is it from?

2$\begingroup$ The answer there gives the names of the authors. You might find the proof in their paper, which you may find by doing a web search. Gerhard "Sounds Like A Duplicate Question" Paseman, 2017.07.01. $\endgroup$– Gerhard PasemanJul 1, 2017 at 20:43

2$\begingroup$ Tried it, dude. Couldn't find it. Which is why I'm asking here. $\endgroup$– GenRincewindJul 1, 2017 at 20:47

5$\begingroup$ @GenRincewind: It is not professional to call your colleagues "dude". BTW the Huxley paper is available for free here: gdz.sub.unigoettingen.de/dms/load/toc/?PPN=PPN356556735 $\endgroup$– GH from MOJul 1, 2017 at 20:48

3$\begingroup$ @GenRincewind: This site is for professional mathematicians. There are many world class scientists here, including senior people, Fields medalists etc. So you cannot be too informal here, just as you don't shout to a university professor that "hey dude, thanks for your comments on my homework!". $\endgroup$– GH from MOJul 1, 2017 at 20:55

2$\begingroup$ Having had a spare moment, I tried typing Harman Baker Pintz into Google's Scholar service. Want to guess what PDF I found as the first listing? Gerhard "It Wasn't Dude Or Dudette" Paseman, 2017.07.01. $\endgroup$– Gerhard PasemanJul 1, 2017 at 21:13
1 Answer
The result is slightly misquoted, it should read $G(X)\leq X^{0.525}$ for $X\geq X_0$. That is, there is no $\log X$ denominator, but $\ll$ can be improved to $\leq$ (when $X$ is sufficiently large). The result appeared in BakerHarmanPintz  The difference between consecutive primes, II (Proc. Lond. Math. Soc. 83 (2001), 532562). If you are interested in the proof, then you need to read it (30 pages).
I recommend you to study the following shorter and less technical paper first: Huxley  On the difference between consecutive primes (Invent. Math. 15 (1972), 164170). It gives $G(X)\ll_\epsilon X^{7/12+\epsilon}$ for any $\epsilon>0$; in fact it gives much more, namely a short interval version of the prime number theorem (unlike BakerHarmanPintz). The prerequisites can be found in IwaniecKowalski's monograph "Analytic number theory".


$\begingroup$ Just to check: G(X) < f(X) denotes that that the maximal prime gap, i.e. the largest gap, below X is less than f(X)? Right? Or am I misunderstanding the notation wholly? $\endgroup$ Jul 1, 2017 at 20:59

$\begingroup$ @GenRincewind: Yes. The precise theorem that BakerHarmanPintz prove is this: for $X\geq X_0$, there is a prime in $[XX^{0.525},X]$. (The condition $X\geq X_0$, i.e. that $X$ is sufficiently large, was missing from my response. Let me add it now.) $\endgroup$ Jul 1, 2017 at 21:01

$\begingroup$ Is there a value for $X_0$ yet or is merely the existence of such a number proven?(I'm still reading through Huxley's paper.) $\endgroup$ Jul 1, 2017 at 21:04

1$\begingroup$ @GenRincewind; The paper only proves the existence of $X_0$. I think though that in the above two papers the constants can be made explicit if one really needs it. $\endgroup$ Jul 1, 2017 at 21:08