{"paper":{"title":"The limiting distribution of Legendre paths","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.PR"],"primary_cat":"math.NT","authors_text":"Ayesha Hussain, Youness Lamzouri","submitted_at":"2023-04-25T17:54:34Z","abstract_excerpt":"Let $p$ be a prime number and $\\left(\\frac{\\cdot}{p}\\right)$ be the Legendre symbol modulo $p$. The \\emph{Legendre path} attached to $p$ is the polygonal path whose vertices are the normalized character sums $\\frac{1}{\\sqrt{p}} \\sum_{n\\leq j} \\left(\\frac{n}{p}\\right)$ for $0\\leq j\\leq p-1$. In this paper, we investigate the distribution of Legendre paths as we vary over the primes $Q\\leq p\\leq 2Q$, when $Q$ is large. Our main result shows that as $Q \\to \\infty$, these paths converge in law, in the space of real-valued continuous functions on $[0, 1]$, to a certain random Fourier series constru"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2304.13025","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2304.13025/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"}