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.)

2more comments