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 ${\sum }_{n\le X}|S_f(n){|}^2$ 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 $|S_f(n){|}^2$ is true on short intervals, and to prove sign change results on $\{S_f(n){\}}_{n\in \mathbb{N}}$.

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
• $p$-adic properties of modular shifted convolution Dirichlet series

Cited by in corpus (7)

• Second Moments in the Generalized Gauss Circle Problem
• Triple Correlation Sums of Coefficients of Cusp Forms
• Sign Changes of Coefficients and Sums of Coefficients of L-Functions
• The Laplace Transform of the Second Moment in the Gauss Circle Problem
• A Shifted Sum for the Congruent Number Problem
• Non-real Poles and Irregularity of Distribution
• Short-Interval Averages of Sums of Fourier Coefficients of Cusp Forms