{"paper":{"title":"The Second Moment of Sums of Coefficients of Cusp Forms","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Alexander Walker, Chan Ieong Kuan, David Lowry-Duda, Thomas A. Hulse","submitted_at":"2015-12-04T01:45:07Z","abstract_excerpt":"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.\n  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 \\ov"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1512.01299","kind":"arxiv","version":4},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}