数学
组合数学
分析化学(期刊)
物理
结晶学
化学
色谱法
出处
期刊:Forum Mathematicum
[De Gruyter]
日期:2023-01-30
卷期号:35 (2): 297-328
被引量:2
标识
DOI:10.1515/forum-2020-0359
摘要
Abstract The purpose of this paper is to obtain asymptotics of shifted sums of Hecke eigenvalue squares on average. We show that for X 2 3 + ϵ < H < X 1 - ϵ {X^{\frac{2}{3}+\epsilon}<H<X^{1-\epsilon}} there are constants B h {B_{h}} such that ∑ X ≤ n ≤ 2 X λ f ( n ) 2 λ f ( n + h ) 2 - B h X = O f , A , ϵ ( X ( log X ) - A ) \sum_{X\leq n\leq 2X}\lambda_{f}(n)^{2}\lambda_{f}(n+h)^{2}-B_{h}X=O_{f,A,% \epsilon}(X(\log X)^{-A}) for all but O f , A , ϵ ( H ( log X ) - 3 A ) {O_{f,A,\epsilon}(H(\log X)^{-3A})} integers h ∈ [ 1 , H ] {h\in[1,H]} where { λ f ( n ) } n ≥ 1 {\{\lambda_{f}(n)\}_{n\geq 1}} 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 {m_{1}} and m 2 {m_{2}} . In order to treat m 2 {m_{2}} , we use the Hecke relations, a bound of Miller to apply some arguments from a paper of Matomäki, Radziwiłł and Tao. In order to treat m 1 {m_{1}} , we apply Parseval’s identity and Gallagher’s lemma.
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