论文标题
Hecke-maass形式的傅立叶系数的第一个同时更改
The first simultaneous sign change for Fourier coefficients of Hecke-Maass forms
论文作者
论文摘要
让$ f $和$ g $为两个hecke-maass cusp，重量为零的零sl_2（\ mathbb z）$，带有laplacian eigenvalues $ \ frac {1} {1} {4}+u^2 $和$ \ \ \ \ \ \ \ frac {1} {1} {1} {4}+v^2 $。然后两个都有真正的傅立叶系数说，$λ_f（n）$和$λ_g（n）$，我们可以正常化$ f $和$ g $，因此$λ_f（1）= 1 =λ_g（1）$。在本文中，我们首先证明了序列$ \ {λ_f（n）λ_g（n）\} _ {n \ in \ mathbb {n}} $具有无限的符号更改。然后，我们根据$ f $和$ g $的拉普拉斯特征值来为同一序列提供第一个负系数的限制。
Let $f$ and $g$ be two Hecke-Maass cusp forms of weight zero for $SL_2(\mathbb Z)$ with Laplacian eigenvalues $\frac{1}{4}+u^2$ and $\frac{1}{4}+v^2$, respectively. Then both have real Fourier coefficients say, $λ_f(n)$ and $λ_g(n)$, and we may normalize $f$ and $g$ so that $λ_f(1)=1=λ_g(1)$. In this article, we first prove that the sequence $\{λ_f(n)λ_g(n)\}_{n \in \mathbb{N}}$ has infinitely many sign changes. Then we derive a bound for the first negative coefficient for the same sequence in terms of the Laplacian eigenvalues of $f$ and $g$.
