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

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 $\sum |S_f(n){|}^2{n}^{-(s+k-1)}$ 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 $X^{\frac{2}{3}}(\log X{)}^{\frac{1}{6}}$.

To Appear in the Journal of Number Theory

References in corpus (4)

• On sums of Fourier coefficients of cusp forms
• Rankin Triple Products and Quantum Chaos
• Subconvexity for Half Integral Weight $L$-functions
• The Second Moment of Sums of Coefficients of Cusp Forms

Cited by in corpus (3)

• Second Moments in the Generalized Gauss Circle Problem
• Sign Changes of Coefficients and Sums of Coefficients of L-Functions
• The Laplace Transform of the Second Moment in the Gauss Circle Problem