{"paper":{"title":"Central Limit Theorems for series of Dirichlet characters","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Andr\\'e LeClair","submitted_at":"2016-12-29T18:30:58Z","abstract_excerpt":"For a given Dirichlet character $\\chi (n) = e^{i \\theta_n}$, we prove central limit theorems for the series $\\sum_{p'} \\cos \\theta_{p'}$ for non-principal characters, and $\\sum_{p' } \\cos (t \\log p')$ for principal characters, where $p'$ are integers based on a variant of Cram\\'er's random model for the primes. For non-principal characters, we use these results to show that the Generalized Riemann Hypothesis for the associated $L$-function is true with probability equal to one. For principal characters we propose how to extend these arguments to $\\Re (s) = t \\to \\infty$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1612.09237","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"}