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Cross-posting from Math.Stackexchange.


You might have read about the fortuitous meeting between Montgomery and Dyson. The background is that the nontrivial zeros of the Riemann zeta function, when normalized to have unit spacing on average, (seem to) have the pair correlation function $1-\mathrm{sinc}^2(x)$, where $\mathrm{sinc}$ is the normalized function $\sin(\pi x)/ (\pi x)$. It's still a conjecture but it has good numerical support.

So what about prime numbers? Let $\Sigma(x,u)$ be the number of pairs of primes $p,q\le x$ which satisfy the inequality $0\le p-q\le \frac{x}{\pi(x)}u$, where $\pi(x)$ is the prime counting function. This inequality is chosen because multiplying primes by $\frac{\pi(x)}{x}$ will ensure the gaps between consecutives is exactly unity (hence they are normalized). Then what might

$$g(u)=\frac{d}{du}\left(\lim_{x\to\infty}\frac{\Sigma(x,u)}{\pi(x)}\right)$$

end up looking like? This basically asks, "what is the density of normalized primes around so-and-so apart from each other?" (You can see the original Montgomery conjecture as equation 12 here. I've adapted it to prime numbers by essentially changing the asymptotic number of zeta zeros to the prime counting function instead. I might be acting presumptuous in assuming the naive replacement will still afford a meaningful answer.)

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Please don't post on MO and M.SE simultaneously. Among other things, it is a bad way of collecting answers in a way useful to other (esp. in the future). –  David Roberts Aug 17 '11 at 21:57
    
@David Roberts: Sorry. I've seen other questions posted on both before and wasn't aware it was looked down upon. –  anon Aug 17 '11 at 22:20
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You may want to take a look at these lectures by Soundararajan: arxiv.org/abs/math/0606408 –  Micah Milinovich Aug 18 '11 at 3:05
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I think whatever result you get follows from the Cramer random model of the primes, or something similar as modified by Maier. Generically, one has that the density of $x$ with $\pi(x+\lambda\log x)−\pi(x)=k$ should be Poisson distributed as $e^{-\lambda}\lambda^k/k!$. This is a nearest-neighbor statistic, and if I am not wrong, the gap distribution should follow. Sorry I do not know off the top of my head or have a reference yet. –  Junkie Aug 18 '11 at 3:13
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Lebeouf does something like this springerlink.com/content/56542774l0171258 –  Junkie Aug 18 '11 at 3:49
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1 Answer

up vote 8 down vote accepted

you are asking for the two-point correlation function of a Poisson process with unit density, which is just unity: $g(u)=1$.

the support for this is about as strong as for the Riemann zeroes: there is extensive numerical evidence but no conclusive theorem; see Soundarajan's 2006 paper cited above, or more recent papers on arXiv:0708.2567 and arXiv:1102.3648

the esssential difference between the function $g_R(u)=1-sinc^2(u)$ for the Riemann zeroes and $g(u)=1$ for the prime numbers, is that the former vanishes $\propto u^2$ for small $u$ ("level repulsion"), while the latter remains constant. This is the difference between the (conjectured) Gaussian unitary ensemble of Riemann zeroes and the Poisson ensemble of prime numbers. For large $u$ all correlations decay and $g_R(u)\rightarrow g(u)$.

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