Higher moments of the pair correlation function for Sato-Tate sequences

TL;DR

This paper investigates higher moments of the pair correlation function for Sato-Tate sequences associated with Hecke angles, providing bounds and convergence results for moments up to the third, advancing understanding of their statistical properties.

Contribution

It extends previous work by deriving bounds for higher moments of the pair correlation function for Hecke angles, including convergence results for the second and third moments.

Findings

Bounds for the r-th power moments of the pair correlation function.

Lower order error terms in second and third moment computations.

Convergence of the second and third moments under certain conditions.

Abstract

In \cite{BS}, Balasubramanyam and the second named author derived the first moment of the pair correlation function for Hecke angles lying in small subintervals of $[0,1]$ upon averaging over large families of Hecke newforms of weight $k$ with respect to $\Gamma_0(N)$. The goal of this article is to study higher moments of this pair correlation function. For an integer $r \geq 2$, we present bounds for its $r$-th power moments. We apply these bounds to record lower order error terms in the computation of the second and third moments. As a result, one can obtain the convergence of the second and third moments of this pair correlation function for suitably small intervals, and under appropriate growth conditions for the size of the families of Hecke newforms.