Pair Dependent Linear Statistics for Circular Beta Ensemble

Abstract: We study limiting distribution of pair counting statistics of the form $ \sum_{1\leq i\neq j\leq N} f(L_N*(\theta_i-\theta_j))$ for the circular $\beta$-ensemble (C$\beta$E) of random matrices for sufficiently smooth test function $f$ and $L_N=O(N).$ For $\beta=2$ and $L_N=N$ our results are inspired by a classical result of Montgomery on pair correlation of zeros of Riemann zeta function.