I'm reading this paper https://arxiv.org/pdf/math/0404114.pdf, and I'm unsure how to interpret the formula in Theorem 2 (1.5), regarding the pair correlation function of the sequence of Farey Fractions. Specifically, I don't understand the $\lambda$ that they used.
In (1.6), they wrote as $\lambda \rightarrow \infty$, $g_2(λ) = 1 + O(λ^{−1})$. So this means the function approaches 1 as $\lambda$ goes to infinity. This seems reasonable as I could see the function asymptotically approaching 1, the problem is I'm not sure what $\lambda$ means in this context.
The only time they mentioned $\lambda$ beforehand is on page 2 when they defined
$R_F^{(v)}(\lambda_1,...,\lambda_{v-1})=2^{-v+1}R_F^{(v)}(\prod_{j=1}^{v-1}[-\lambda_j,\lambda_j]).$
My guess right now is that $\lambda$ refers to the interval for which the correlation measures are taken. Thus, as the interval of normalized Farey Fractions become arbitrarily large, the function converges to 1. Is that reasonable? Then $\lambda$ would represent a box in $\mathbb{R}^{v-1}$ for $v \geq3$.
Could someone give me some clarity on what this $\lambda$ represents and how this would change for $v\geq 3$? Thanks a lot.
It might be a better idea not to look at the (free) parameter $\lambda$, but instead to look at $g_2$. On page 2, $g_2$ is defined as a function such that the $2$-level correlation measure is given by
$$ \mathcal{R}^{2}(\mathcal{B}) = \int_{\mathcal{B}} g_2(x_1) dx_1.$$
So $\lambda$ (in their Theorem 2) is just a number. I suppose they could have called it $x_1$, but that seems pretty odd.
Note that this is strongly analogous to the pair correlation function for $\zeta(s)$.
An analogous function for higher level correlation would necessarily be higher-dimensional. This is sometimes studied along with random-matrix theory, and called "$n$-level density". See for example this preprint by Chandee and Lee.