{"paper":{"title":"An asymptotic expansion for the generalised quadratic Gauss sum revisited","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"R B Paris","submitted_at":"2014-03-31T12:41:43Z","abstract_excerpt":"An asymptotic expansion for the generalised quadratic Gauss sum $$S_N(x,\\theta)=\\sum_{j=1}^{N} \\exp (\\pi ixj^2+2\\pi ij\\theta),$$ where $x$, $\\theta$ are real and $N$ is a positive integer, is obtained as $x\\rightarrow 0$ and $N\\rightarrow\\infty$ such that $Nx$ is finite. The form of this expansion holds for all values of $Nx+\\theta$ and, in particular, in the neighbourhood of integer values of $Nx+\\theta$. A simple bound for the remainder in the expansion is derived. Numerical results are presented to demonstrate the accuracy of the expansion and the sharpness of the bound."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1403.7973","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"}