Iwaniec, Luo, and Sarnak proposed Hypothesis S and its generalization which predicts non-trivial bounds for a smooth sum of the product of an arithmetic sequence $\{a_n\}$ and a fractional exponential function. When $a_n$ is the Fourier coefficient ${\lambda }_f(n)$ of a fixed holomorphic cusp form f, however, a resonance phenomenon prohibits any improvement of the bound beyond a barrier. It is believed that this resonance barrier could be overcome when the weight k of f tends to infinity. The present paper is a first step toward this goal by proving non-trivial bounds for this sum when k and the summation length X both tend to infinity. No such non-trivial bounds are previously known if the form f is allowed to move. Similar bounds are also proved for linear phases and for Maass forms. The main technology is improved large sieve inequalities over a short interval.

Iwaniec、Luo 和 Sarnak 提出了假设 S 及其泛化，它预测了等差数列乘积的平滑和的非平凡界限$\{{\mathrm{一种}}_n\}$和分数指数函数。什么时候${\mathrm{一种}}_n$是傅立叶系数${\lambda }_F(n)$然而，对于固定的全纯尖点形式f，共振现象禁止任何超越障碍的界限的改进。相信当f的权重k趋于无穷大时，这种共振势垒可以被克服。本文通过证明当k和总和长度X都趋于无穷大时该总和的非平凡界限是朝着这个目标迈出的第一步。如果允许形式f移动，则以前不知道这样的非平凡界限。对于线性相和 Maass 形式也证明了类似的界限。主要技术是在短时间内改进大筛子不等式。