论文标题
几乎所有短时间间隔
On Hecke eigenvalues of cusp forms in almost all short intervals
论文作者
论文摘要
令$ψ$成为一个函数，以至于$ψ（x）\ rightarrow \ ins in his $ x \ rightarrow \ infty。$ ling $λ_{f}（n）$为$ n $ -n $ -th hecke eigenvalue for $ s $ f $ for $ sl for $ sl（ $ h（x）$使得$（\ log x）^{2-2α} \ ll H（x）= o（x），$ $ $ $ \ sum_ {n = x}^}^{x+h（x）} |λ__{f} \ ll_ {f} h（x）ψ（x）（\ log x）^{α-1} $$除$ o_ {f}（xψ（xψ（x x）^{ - 2}）$许多整数$ x \ in [x，2x-h（x）]，$ a $ las $α$的平均值$ | pr | las $ |λ_{我们将其概括为$ |λ_{f}（n）|^{2^{k}} $ for $ k \ in \ mathbb {z^{+}}。
Let $ψ$ be a function such that $ψ(x) \rightarrow \infty$ as $x \rightarrow \infty.$ Let $λ_{f}(n)$ be the $n$-th Hecke eigenvalue of a fixed holomorphic cusp form $f$ for $SL(2,\mathbb{Z}).$ We show that for any real valued function $h(x)$ such that $(\log X)^{2-2α} \ll h(X) =o(X),$ $$\sum_{n=x}^{x+h(X)} |λ_{f}(n)| \ll_{f} h(X)ψ(X)(\log X)^{α-1}$$ for all but $O_{f}( Xψ(X)^{-2})$ many integers $x\in [X,2X-h(X)],$ in which $α$ is the average value of $|λ_{f}(p)|$ over primes. We generalize this for $|λ_{f}(n)|^{2^{k}}$ for $k \in \mathbb{Z^{+}}.$
