{"paper":{"title":"A refinement of the Burgess bound for character sums","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Bryce Kerr, Igor E. Shparlinski, Kam Hung Yau","submitted_at":"2017-11-28T22:03:40Z","abstract_excerpt":"In this paper we give a refinement of the bound of D. A. Burgess for multiplicative character sums modulo a prime number $q$. This continues a series of previous logarithmic improvements, which are mostly due to H. Iwaniec and E. Kowalski. In particular, for any nontrivial multiplicative character $\\chi$ modulo a prime $q$ and any integer $r\\ge 2$, we show that $$ \\sum_{M<n\\le M+N}\\chi(n) = O\\left( N^{1-1/r}q^{(r+1)/4r^2}(\\log q)^{1/4r}\\right), $$ which sharpens previous results by a factor $(\\log q)^{1/4r}$. Our improvement comes from averaging over numbers with no small prime factors rather "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1711.10582","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"}