On the higher moments of the error term in the Rankin-Selberg problem
Jing Huang, Yoshio Tanigawa, Wenguang Zhai, Deyu Zhang
TL;DR
The paper advances the Rankin-Selberg program by deriving asymptotic formulas for the third, fourth, and fifth power moments of the error term $\Delta_1(x;\varphi)$ in the Rankin-Selberg problem. A Voronoi-type decomposition $\Delta_1(x;\varphi) = (2\pi)^{-2}\mathcal{R}_1(x;N) + \mathcal{R}_2(x;y)$ with a truncation parameter $y$ is employed to isolate a main term, with $\mathcal{R}_1^k$ providing the principal contribution and cross-terms controlled via oscillatory- and diophantine-type bounds. For each $k\in\{3,4,5\}$, the authors obtain an explicit main term involving $B_k(c)$ and the integral of $x^{9k/8}$, together with refined error terms $O(T^{1+9k/8-\delta_k+\varepsilon})$, where $\delta_3=3/62$, $\delta_4=3/256$, $\delta_5=1/680$, after optimizing the truncation parameter $y$. These results improve prior work by Tanigawa, Zhai, and Zhang, and deepen the understanding of high-moment behavior in the Rankin-Selberg setting using Voronoi-type methods and oscillatory-analysis techniques.
Abstract
Let $Δ_1(x;\varphi)$ denote the error term in the classical Rankin-Selberg problem. In this paper, we consider the higher power moments of $Δ_1(x;\varphi)$ and derive the asymptotic formulas for 3-rd, 4-th and 5-th power moments, which improve the previous results.