Average behaviour of Hecke eigenvalues over certain polynomial

TL;DR

This paper studies the average behavior of normalized Hecke eigenvalues over specific polynomial values, providing asymptotic estimates for their power moments when summed over square-free integers.

Contribution

It establishes new asymptotic formulas for the moments of Hecke eigenvalues over polynomial values involving sums of triangular numbers.

Findings

Asymptotic formulas for power moments of Hecke eigenvalues

Results for sums over square-free integers

Extension to polynomial values involving triangular numbers

Abstract

In the article, we investigate the average behaviour of normalised Hecke eigenvalues over certain polynomials and establish an estimate for the power moments of the normalised Hecke eigenvalues of a normalised Hecke eigenform of weight $k \ge 2$ for the full modular group $SL_2(\mathbb{Z})$ over certain polynomial, given by a sum of triangular numbers with certain positive coefficients. More precisely, for each $r \in \mathbb{N}$, we obtain an asymptotic for the following sum \begin{equation*} \begin{split} \displaystyle{\sideset{}{^{\flat }}\sum_{ \alpha(\underline{x}))+1 \le X \atop \underline{x} \in {\mathbb Z}^{4}} } \lambda_{f}^{r}(\alpha(\underline{x})+1) , \\ \end{split} \end{equation*} where $\displaystyle{\sideset{}{^{\flat }}\sum}$ means that the sum runs over the square-free positive integers, and $\lambda_{f} (n)$ is the normalised $n^{\rm th}$-Hecke eigenvalue of a normalised Hecke eigenform $f \in S_{k}(SL_2(\mathbb{Z}))$, and $\alpha(\underline{x}) = \frac{1}{2} \left( x_{1}^{2}+ x_{1} + x_{2}^{2} + x_{2} + 2 ( x_{3}^{2} + x_{3}) + 4 (x_{4}^{2} + x_{4}) \right) \in {\mathbb Q}[x_{1},x_{2},x_{3},x_{4}] $ is a polynomial, and $\underline{x} = (x_{1},x_{2},x_{3},x_{4}) \in {\mathbb Z}^{4}$.