论文标题
在涉及Maass尖的傅里叶系数的一些估计中
On some estimates involving Fourier coefficients of Maass cusp forms
论文作者
论文摘要
令$ f $为$ \ rm sl_2（\ mathbb {z}）$的hecke-maass cusp表格，带有laplace eigenvalue $λ_f（δ）= 1/4+μ^2 $，让$λ_f（n）$是其$ n $ n $ n $ th n $ th normanizip fourigized fourimized fourimized fourimizip fourimizip fourimizip fourizizip fourimized fourier ceeff。事实证明，在$α，β\ in \ mathbb {r} $，$$ \ sum_ {n \ leq x}λ_f（n）e \ left（αn^2+βn\ right） $$隐含常数仅取决于$ \ varepsilon $。我们还考虑了$λ_f（n）$的求和函数，在Ramanujan猜想下，我们能够证明$$ \ sum_ {n \ leq x}λ_f（n）\ ll x^{1/3+\ \ \ \ \ varepsilon}λ_f（Δ） $ \ varepsilon $。
Let $f$ be a Hecke-Maass cusp form for $\rm SL_2(\mathbb{Z})$ with Laplace eigenvalue $λ_f(Δ)=1/4+μ^2$ and let $λ_f(n)$ be its $n$-th normalized Fourier coefficient. It is proved that, uniformly in $α, β\in \mathbb{R}$, $$ \sum_{n \leq X}λ_f(n)e\left(αn^2+βn\right) \ll X^{7/8+\varepsilon}λ_f(Δ)^{1/2+\varepsilon}, $$ where the implied constant depends only on $\varepsilon$. We also consider the summation function of $λ_f(n)$ and under the Ramanujan conjecture we are able to prove $$ \sum_{n \leq X}λ_f(n)\ll X^{1/3+\varepsilon}λ_f(Δ)^{4/9+\varepsilon} $$ with the implied constant depending only on $\varepsilon$.
