The merit of a prime gap equals $(p_{n+1}-p_n)/\ln p_n$. One can interrogate the statistics of merit by first restricting $n<M$ for some $M$, and then letting $M$ approach $\infty$. The very definition got rigged to make the mean of merit equal 1 in the limit. What about higher moments? Do we have theorems, standard conjectures, or even just numerical evidence concerning their values or their mere existence?
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asked May 7, 2018 at 6:18
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3$\begingroup$ I don't think "merit" is a term commonly used for this concept; "normalized prime gap" is common and quite descriptive. $\endgroup$– Greg MartinCommented May 7, 2018 at 16:55
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$\begingroup$ I picked up the term from en.wikipedia.org/wiki/Prime_gap $\endgroup$– David FeldmanCommented May 7, 2018 at 17:26
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$\begingroup$ There is also the Cramér–Shanks–Granville ratio, a candidate for a different normalized prime gap. $\endgroup$– David FeldmanCommented May 7, 2018 at 17:27
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$\begingroup$ Maybe the quantity defined by $ \log g_{n}/\log\log n $ where $ g_{n}=p_{n+1}-p_{n} $ can be of interest too. $\endgroup$– Sylvain JULIENCommented May 7, 2018 at 20:02
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$\begingroup$ @sylvain-julien Or perhaps $g_n/(\ln n)^2$. The heuristic: if x has primeness probability $1/\ln x$, then $\ln x$ consecutive composites starting at x should happen with probability about $(1 - 1/\ln x)^{\ln x}$, so $1/e$ in the limit. $c(\ln x)^2$ consecutive composites would occur with probability $1/e^{c\ln x}= 1/x^c$ making the expected number of occurrences finite for $c>1$. Thus one might expect a bound for my proposed quantity and even ask whether a particular gap realizes a maximum value. $\endgroup$– David FeldmanCommented May 10, 2018 at 0:49
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1 Answer
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The standard conjecture is that the statistics of $(p_{n+1}-p_n)/\ln n$ are, in the limit, precisely an exponential distribution with parameter $1$; one can recover conjectures about individual moments from this, although evaluating the moments is the natural way to approach the conjecture. This is the natural heuristic that follows from the model that every integer $m$ is prime with probability $1/\ln m$. I believe this conjecture was proved conditionally by Gallagher (in Mathematika in 1976), assuming a strong form of the prime $k$-tuples conjecture.
answered May 7, 2018 at 7:50
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1$\begingroup$ Moreover, a similar Poisson distribution should hold for successive twin primes, successive primes of the from $n^2+1$, etc, etc... $\endgroup$ Commented May 7, 2018 at 17:49
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1$\begingroup$ One must be a tiny bit careful: Indeed the distribution should be Poisson, but the moments could blow up if there is an unusually large prime gap. Gallagher's result, just to be precise, is based on the moments of $\pi(x+\lambda \log x)-\pi(x)$, which has the advantage of not being able to blow up! $\endgroup$– LuciaCommented May 7, 2018 at 21:20
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$\begingroup$ Thanks Lucia for pointing out that pitfall when translating Gallagher's work to this situation! $\endgroup$ Commented May 7, 2018 at 22:33
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$\begingroup$ Greg Martin -- Actually I'm confused. A Poisson distribution is a distribution on the counting numbers. If $π(x+\ln x)−π(x)$ is Poisson, wouldn't that mean that an exponential distribution governs merit? So the moments should be 1,2,6,24,...?? (Which doesn't jive with my calculations by the way.) $\endgroup$ Commented May 8, 2018 at 19:17
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1$\begingroup$ I accepted this answer...but don't you mean an exponential distribution?? $\endgroup$ Commented May 10, 2018 at 18:10