On the asymptotics of the shifted sums of Hecke eigenvalue squares

TL;DR

This paper derives asymptotic formulas for shifted sums of squares of Hecke eigenvalues on average, using the Hardy-Littlewood circle method and advanced analytic techniques, with implications for understanding automorphic forms.

Contribution

It introduces a novel approach to asymptotics of shifted sums of Hecke eigenvalues using the circle method and Hecke relations, improving previous bounds and methods.

Findings

Asymptotic formulas hold for most shifts within specified ranges.

Method combines circle method with Hecke relations and bounds from Miller.

Results apply to normalized Hecke eigenvalues of holomorphic cusp forms.

Abstract

The purpose of this paper is to obtain asymptotics of shifted sums of Hecke eigenvalue squares on average. We show that for $X^{\frac{2}{3}+\epsilon} < H <X^{1-\epsilon},$ there are constants $B_{h}$ such that $$ \sum_{X\leq n \leq 2X} \lambda_{f}(n)^{2}\lambda_{f}(n+h)^{2}-B_{h}X=O_{f,A,\epsilon}\big(X (\log X)^{-A}\big)$$ for all but $O_{f,A,\epsilon}\big(H(\log X)^{-3A}\big)$ integers $h \in [1,H]$ where $\{\lambda_{f}(n)\}_{n\geq1}$ are normalized Hecke eigenvalues of a fixed holomorphic cusp form $f.$ Our method is based on the Hardy-Littlewood circle method. We divide the minor arcs into two parts $m_{1}$ and $m_{2}.$ In order to treat $m_{2},$ we use the Hecke relations, a bound of Miller to apply some arguments from a paper of Matom\"{a}ki, Radziwill and Tao. We apply Parseval's identity and Gallagher's lemma so as to treat $m_{1}.$