{"paper":{"title":"Large character sums: Burgess's theorem and zeros of $L$-functions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Andrew Granville, Kannan Soundararajan","submitted_at":"2015-01-08T11:47:04Z","abstract_excerpt":"We study the conjecture that $\\sum_{n\\leq x} \\chi(n)=o(x)$ for any primitive Dirichlet character $\\chi \\pmod q$ with $x\\geq q^\\epsilon$, which is known to be true if the Riemann Hypothesis holds for $L(s,\\chi)$. We show that it holds under the weaker assumption that `$100\\%$' of the zeros of $L(s,\\chi)$ up to height $\\tfrac 14$ lie on the critical line; and establish various other consequences of having large character sums."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1501.01804","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"}