Short-Interval Averages of Sums of Fourier Coefficients of Cusp Forms

Thomas A. Hulse, Chan Ieong Kuan, David Lowry-Duda, Alexander Walker

Published 2015-12-17Version 1

Let $f$ be a weight $k$ holomorphic cusp form of level one, and let $S_f(n)$ denote the sum of the first $n$ Fourier coefficients of $f$. In analogy with Dirichlet's divisor problem, it is conjectured that $S_f(X) \ll X^{\frac{k-1}{2} + \frac{1}{4} + \epsilon}$. Understanding and bounding $S_f(X)$ has been a very active area of research. The current best bound for individual $S_f(X)$ is $S_f(X) \ll X^{\frac{k-1}{2} + \frac{1}{3}} (\log X)^{-0.1185}$ from Wu. Chandrasekharan and Narasimhan showed that the Classical Conjecture for $S_f(X)$ holds on average over intervals of length $X$. Jutila improved this result to show that the Classical Conjecture for $S_f(X)$ holds on average over short intervals of length $X^{\frac{3}{4} + \epsilon}$. Building on the results and analytic information about $\sum \lvert S_f(n) \rvert^2 n^{-(s + k - 1)}$ from our recent paper, we further improve these results to show that the Classical Conjecture for $S_f(X)$ holds on average over short intervals of length approximately $X^{\frac{2}{3} + \epsilon}$.