The Second Moment of Sums of Coefficients of Cusp Forms

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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 Ivic [HI], building on Chandrasekharan and Narasimhan [CN], proved asymptotics for ∑ₙ ≤ ₓ Sf(n) ² and proved that the Classical Conjecture, that Sf(X) ≪ Xᵏ⁻¹/² ⁺ ¹/⁴ ⁺ 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) = ∑ₙ Sf(n)Sg(n) n⁻⁽ˢ ⁺ ᵏ ⁻ ¹⁾. Using these meromorphic continuations, we prove asymptotics for the smoothed second moment sums ∑ Sf(n)Sg(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× 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 Sf(n) ² is true on short intervals, and to prove sign change results on {Sf(n)}ₙ ∈ ₙ.