{"paper":{"title":"Fluctuations in the distribution of Hecke eigenvalues about the Sato-Tate measure","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Kaneenika Sinha, Neha Prabhu","submitted_at":"2017-05-11T11:36:00Z","abstract_excerpt":"We study fluctuations in the distribution of families of $p$-th Fourier coefficients $a_f(p)$ of normalised holomorphic Hecke eigenforms $f$ of weight $k$ with respect to $SL_2(\\mathbb{Z})$ as $k \\to \\infty$ and primes $p \\to \\infty.$ These families are known to be equidistributed with respect to the Sato-Tate measure. We consider a fixed interval $I \\subset [-2,2]$ and derive the variance of the number of $a_f(p)$'s lying in $I$ as $p \\to \\infty$ and $k \\to \\infty$ (at a suitably fast rate). The number of $a_f(p)$'s lying in $I$ is shown to asymptotically follow a Gaussian distribution when a"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1705.04115","kind":"arxiv","version":2},"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"}