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

TL;DR

This paper improves bounds on the average behavior of sums of Fourier coefficients of cusp forms over short intervals, advancing understanding of their growth and distribution in analogy with classical divisor problems.

Contribution

It extends previous results by demonstrating that the classical conjecture holds on average over shorter intervals of length approximately X^{2/3} with logarithmic factors.

Findings

Classical conjecture holds on average over intervals of length X^{2/3} (log X)^{1/6}

Improves previous bounds for short interval averages of Fourier coefficient sums

Builds on recent analytic techniques and prior work to refine interval length estimates

Abstract

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 work, we further improve these results to show that the Classical Conjecture for $S_f(X)$ holds on average over short intervals of length $X^{\frac{2}{3}}(\log X)^{\frac{1}{6}}$.