Shifted convolution sums related to Hecke-Maass forms
Shifted convolution sums related to Hecke-Maass forms
复制标题
与 Hecke-Maass 形式相关的移位卷积和
DOI：
10.1007/s11139-019-00244-y
复制
发表时间：
2021
影响因子：
0.7
通讯作者：
Wu Jie
中科院分区：
文献类型：
--
作者：
Tang Hengcai;Wu Jie
设一个本原Hecke-Maass尖点由Laplace本征值构成。用它们表示与的Dirichlet展开式的第n个系数有关的第n个对称幂L-函数。对于任何非零整数，我们证明了$$\Begin{Aligned}\sum_{n\leqslant x}\Left|\lambda_{\Phi}(N)\lambda_{\Phi}(n+\ell)\right|\ll_{\Phi，\ell}\frac{x}{(\logx)^{0.187}}\qquad(x\geqslant 3)。这改进了霍洛温斯基的相应结果，它需要0.187的位置。为了所有人。进一步假设AND是自同构尖锥形的，我们得到了对称正方形情形的一个条件推广：$$\Begin{Align}\sum_{n\leqslant x}\Left|\lambda_{\mathm{sym}^2\Phi}(N)\lambda_{\mathm{sym}^2\Phi}(n+\ell)\right|\ll_{\Phi，\ell}\frac{x}{(\logx)^{0.196}}\qquad(x\geqslant 3)。\结束{已对齐}$$
Letbe a primitive Hecke–Maass cusp forms with Laplace eigenvalue. Denote bythem-th symmetric powerL-function associated toand bythen-th coefficient of the Dirichlet expansion of. For any nonzero integerwe prove $$\begin{aligned} \sum _{n\leqslant x} \left| \lambda _{\phi }(n)\lambda _{\phi }(n+\ell )\right| \ll _{\phi , \ell } \frac{x}{(\log x)^{0.187}} \qquad (x\geqslant 3). \end{aligned}$$This improves Holowinsky’s corresponding result, which requiresin place of 0.187. for all. Further assuming thatandare automorphic cuspidal, we obtain a conditional generalization to the symmetric square case: $$\begin{aligned} \sum _{n\leqslant x} \left| \lambda _{\mathrm{sym}^2\phi }(n)\lambda _{\mathrm{sym}^2\phi }(n+\ell )\right| \ll _{\phi , \ell } \frac{x}{(\log x)^{0.196}} \qquad (x\geqslant 3). \end{aligned}$$