On Hecke eigenvalues of cusp forms in almost all short intervals

Abstract

Let $\psi$ be a function such that $\psi (x)\to \infty$ as $x\to \infty .$ Let ${\lambda }_f(n)$ 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 $(\log X{)}^{2-2\alpha }\ll h(X)=o(X),$ $$\sum _{n=x}^{x+h(X)}|{\lambda }_f(n)|{\ll }_{f}h(X)\psi (X)(\log X{)}^{\alpha -1}$$ for all but $O_f(X\psi (X{)}^{-2})$ many integers $x\in [X,2X-h(X)],$ in which $\alpha$ is the average value of $|{\lambda }_f(p)|$ over primes. We generalize this for $|{\lambda }_f(n){|}^{2^k}$ for $k\in {\mathbb{Z}}^{+}.$

arXiv commentTo appear in International Journal of Number Theory