Montgomery's pair correlation conjecture states that the distribution of the pair correlations of the zeroes of the Riemann zeta function (normalized to have average spacing 1) is given by the function
$$
g(u) = 1 - \left(\frac{\sin(\pi u)}{\pi u}\right)^2 + \delta(u).
$$
See <https://en.wikipedia.org/wiki/Pair_correlation_conjecture> for some basic information. Famously, this distribution also appears in random matrix theory.


There is also a conjecture for the distribution of the nearest neighbor spacings of the normalized zeroes, which is conjectured to be given by 
$$
f(u) = \frac{32}{\pi^2} u^2 \exp\left(-\frac{4}{\pi} u^2 \right)
$$


Question: What is the conjectured distribution of the second-nearest neighbors? Can it be obtained by assuming that consecutive spacings are stochastically *independent*? - if "yes", then we could obtain the conjectured distribution $f\_2(u)$ of the spacings of the second-nearest neighbors by convolution, that is, by the formula $f\_2(u) = (f\*f) (u)$. Does it work this way?


Furtheremore, writing $f^{\*k}$ for $\underbrace{f\* \dots \* f}\_{k \textrm{ times}}$ - then if all consecutive spacings were conjectured to be independent, we should have
$$
g(u) = \delta(u) + \sum\_{k=1}^\infty f^{\*k}(u).
$$
Is this last equality true? (I don't know how to check it.)



Riemann zeta function: pair correlations vs. neighbor spacings
This next-nearest-neighbor distribution of the Riemann zero's is addressed in [Mehta's book on random-matrix theory.](https://books.google.nl/books/about/Random_Matrices.html?id=Kp3Nx03_gMwC&redir_esc=y) It is well reproduced by that of the Gaussian Unitary Ensemble (GUE), compare black curve and black data points:


![](https://ilorentz.org/beenakker/MO/nnn_GUE.png)


I added the red curve in the plot, being the convolution of the nearest-spacing distribution in the GUE:


$$p(s)=\int\_0^s f(u)f(s-u)du,\;\;\text{with}\;\;f(u) = \frac{32}{\pi^2} u^2 \exp\left(-\frac{4}{\pi} u^2 \right)$$ 


As you can see, it is quite different, so this "independent spacing" assumption is not reliable.


All of this is for complex matrices (GUE). For real symmetric matrices (GOE) the distribution of next-nearest-neighbor spacings is given by the distribution of the nearest-neighbor spacings of quaternion Hermitian matrices (GSE).


