Cross-posting from [Math.Stackexchange](https://math.stackexchange.com/questions/58168/what-might-the-normalized-pair-correlation-function-of-prime-numbers-look-like).




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You might have read about the [fortuitous meeting between Montgomery and Dyson](http://www.americanscientist.org/issues/pub/the-spectrum-of-riemannium). 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](http://www.jstor.org/pss/2007890).


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](http://www-personal.umich.edu/~hlm/paircor1.pdf). 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.)



What might the (normalized) pair correlation function of prime numbers look like?
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 $g\_R(u)\rightarrow g(u)$.


