{"paper":{"title":"Quadratic nonresidues below the Burgess bound","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Victor Z. Guo, William D. Banks","submitted_at":"2015-11-17T19:51:10Z","abstract_excerpt":"For any odd prime number $p$, let $(\\cdot|p)$ be the Legendre symbol, and let $n_1(p)<n_2(p)<\\cdots$ be the sequence of positive nonresidues modulo $p$, i.e., $(n_k|p)=-1$ for each $k$. In 1957, Burgess showed that the upper bound $n_1(p)\\ll_\\epsilon p^{(4\\sqrt{e})^{-1}+\\epsilon}$ holds for any fixed $\\epsilon>0$. In this paper, we prove that the stronger bound $$ n_k(p)\\ll p^{(4\\sqrt{e})^{-1}}\\exp\\big(\\sqrt{e^{-1}\\log p\\log\\log p}\\,\\big) $$ holds for all odd primes $p$, where the implied constant is absolute, provided that $$ k\\le p^{(8\\sqrt{e})^{-1}} \\exp\\big(\\tfrac12\\sqrt{e^{-1}\\log p\\log\\l"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1511.05523","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":""},"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"}