{"paper":{"title":"The typical size of character and zeta sums is $o(\\sqrt{x})$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Adam J. Harper","submitted_at":"2023-01-11T10:28:14Z","abstract_excerpt":"We prove conjecturally sharp upper bounds for the Dirichlet character moments $\\frac{1}{r-1} \\sum_{\\chi \\; \\text{mod} \\; r} |\\sum_{n \\leq x} \\chi(n)|^{2q}$, where $r$ is a large prime, $1 \\leq x \\leq r$, and $0 \\leq q \\leq 1$ is real. In particular, if both $x$ and $r/x$ tend to infinity with $r$ then $\\frac{1}{r-1} \\sum_{\\chi \\; \\text{mod} \\; r} |\\sum_{n \\leq x} \\chi(n)| = o(\\sqrt{x})$, and so the sums $\\sum_{n \\leq x} \\chi(n)$ typically exhibit \"better than squareroot cancellation\". We prove analogous better than squareroot bounds for the moments $\\frac{1}{T} \\int_{0}^{T} |\\sum_{n \\leq x} n^"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2301.04390","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2301.04390/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}