The second moment of sums of coefficients of cusp forms

Abstract

Let f and g be weight k holomorphic cusp forms and let Sf(n) and Sg(n) denote the sums of their first n Fourier coefficients. Hafner and Ivić [9] proved asymptotics for ∑n≤X|Sf(n)|2 and proved that the Classical Conjecture, that Sf(X)≪Xk−12+14+ϵ, holds on average over long intervals.In this paper, we introduce and obtain meromorphic continuations for the Dirichlet series D(s,Sf×Sg)=∑Sf(n)Sg(n)‾×n−(s+k−1) and D(s,Sf×Sg‾)=∑nSf(n)Sg(n)n−(s+k−1). We then prove asymptotics for the smoothed second moment sums ∑Sf(n)Sg(n)‾e−n/X, giving a smoothed generalization of [9]. We also attain asymptotics for analogous sums of normalized Fourier coefficients. Our methodology extends to a wide variety of weights and levels, and comparison with [4] indicates very general cancellation between the Rankin–Selberg L-function L(s,f×g) and 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 |Sf(n)|2 is true on short intervals, and to prove sign change results on {Sf(n)}n∈N.