{"paper":{"title":"A lower bound on high moments of character sums","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Barnab\\'as Szab\\'o","submitted_at":"2024-09-20T11:59:43Z","abstract_excerpt":"For any real $k\\geq 2$ and large prime $q$, we prove a lower bound on the $2k$-th moment of the Dirichlet character sum\n  \\begin{equation*}\n  \\frac{1}{\\phi(q)} \\sum_{\\substack{\\chi \\text{ mod }q\\\\ \\chi\\neq \\chi_0}} \\Big| \\sum_{n\\leq x} \\chi(n)\\Big|^{2k}, \\end{equation*} where $1\\leq x\\leq q$, and $\\chi$ is summed over the set of non-trivial Dirichlet characters mod $q$. Our bound is known to be optimal up to a constant factor under the Generalised Riemann Hypothesis. We also get a sharp lower bound on moments of theta functions using the same method."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2409.13436","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/2409.13436/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"}