论文标题
在$ \ mathrm {gl} _3 $上的hecke-maass尖forms的傅立叶系数的迹象
On Signs of Fourier Coefficients of Hecke-Maass Cusp Forms on $\mathrm{GL}_3$
论文作者
论文摘要
我们考虑了组$ \ mathrm {sl} _3（\ Mathbb z）$的Hecke-maass尖form的傅立叶系数的符号更改。当基本形式是自动对齐的时，我们表明有$ \ gg_ \ varepsilon x^{5/6- \ varepsilon} $符号变化的系数$ \ {a（m，1）_ {m，1）_ {m \ \ leq x} $，并且对许多自我自我征兆的比例更改为正面形式。关于符号更改的正面比例的类似结果，对于通用$ \ mathrm {gl} _3 $ cusp forme的实值系数$ a（m，m）$也可以得出，这一结果基于新的有效的sato-tate类型定理，用于$ \ m m i \ nathrm {gl} _3} _3 $ cusp cusp fore。另外，在Ramanujan-Petersson的猜想下研究了傅立叶系数的不变。
We consider sign changes of Fourier coefficients of Hecke-Maass cusp forms for the group $\mathrm{SL}_3(\mathbb Z)$. When the underlying form is self-dual, we show that there are $\gg_\varepsilon X^{5/6-\varepsilon}$ sign changes among the coefficients $\{A(m,1)\}_{m\leq X}$ and that there is a positive proportion of sign changes for many self-dual forms. Similar result concerning the positive proportion of sign changes also hold for the real-valued coefficients $A(m,m)$ for generic $\mathrm{GL}_3$ cusp forms, a result which is based on a new effective Sato-Tate type theorem for a family of $\mathrm{GL}_3$ cusp forms we establish. In addition, non-vanishing of the Fourier coefficients is studied under the Ramanujan-Petersson conjecture.
