论文标题
在移动的hecke特征值平方的渐近级上
On the asymptotics of the shifted sums of Hecke eigenvalue squares
论文作者
论文摘要
本文的目的是平均获得Hecke特征值正方形的转移总和的渐近性。我们表明，对于$ x^{\ frac {2} {3}+ε} <h <x^{1-ε}，$有常数$ b_ {h} $ λ_{f}（n）^{2}λ_{f}（n+h）^{2} -B_ { X)^{-3A}\big)$ integers $h \in [1,H]$ where $\{λ_{f}(n)\}_{n\geq1}$ are normalized Hecke eigenvalues of a fixed holomorphic cusp form $f.$ Our method is based on the Hardy-Littlewood circle method.我们将小弧分为两个部分$ m_ {1} $和$ m_ {2}。$为了对待$ m_ {2}，$，我们使用hecke关系，米勒（Miller）的界限将一些参数应用于matomäki，radziwill和tao的论文中。我们应用Parseval的身份和Gallagher的引理，以治疗$ M_ {1}。
The purpose of this paper is to obtain asymptotics of shifted sums of Hecke eigenvalue squares on average. We show that for $X^{\frac{2}{3}+ε} < H <X^{1-ε},$ there are constants $B_{h}$ such that $$ \sum_{X\leq n \leq 2X} λ_{f}(n)^{2}λ_{f}(n+h)^{2}-B_{h}X=O_{f,A,ε}\big(X (\log X)^{-A}\big)$$ for all but $O_{f,A,ε}\big(H(\log X)^{-3A}\big)$ integers $h \in [1,H]$ where $\{λ_{f}(n)\}_{n\geq1}$ 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 Matomäki, Radziwill and Tao. We apply Parseval's identity and Gallagher's lemma so as to treat $m_{1}.$
