The Second Moment of Sums of Coefficients of Cusp Forms

Abstract

Let $f$ and $g$ be weight $k$ holomorphic cusp forms and let $S_f(n)$ and $S_g(n)$ denote the sums of their first $n$ Fourier coefficients. Hafner and Ivic [HI], building on Chandrasekharan and Narasimhan [CN], proved asymptotics for $\sum_{n \leq X} \lvert S_f(n) \rvert^2$ and proved that the Classical Conjecture, that $S_f(X) \ll X^{\frac{k-1}{2} + \frac{1}{4} + \epsilon}$, holds on average over long intervals. In this paper, we introduce and obtain meromorphic continuations for the Dirichlet series $D(s, S_f \times S_g) = \sum S_f(n)\overline{S_g(n)} n^{-(s+k-1)}$ and $D(s, S_f \times \overline{S_g}) = \sum_n S_f(n)S_g(n) n^{-(s + k - 1)}$. Using these meromorphic continuations, we prove asymptotics for the smoothed second moment sums $\sum S_f(n)\overline{S_g(n)} e^{-n/X}$, proving a smoothed generalization of [HI]. We also attain asymptotics for analogous smoothed second moment sums of normalized Fourier coefficients, proving smoothed generalizations of what would be attainable from [CN]. Our methodology extends to a wide variety of weights and levels, and comparison with [CN] indicates very general cancellation between the Rankin-Selberg $L$-function $L(s, f\times g)$ and shifted convolution sums of the coefficients of $f$ and $g$. In forthcoming works, the authors apply the results of this paper to prove the Classical Conjecture on $\lvert S_f(n) \rvert^2$ is true on short intervals, and to prove sign change results on $\{S_f(n)\}_{n \in \mathbb{N}}$.