{"paper":{"title":"Reciprocity Theorems for Bettin--Conrey Sums","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Abdelmejid Bayad, Juan S. Auli, Matthias Beck","submitted_at":"2016-01-25T22:40:06Z","abstract_excerpt":"Recent work of Bettin and Conrey on the period functions of Eisenstein series naturally gave rise to the Dedekind-like sum \\[\n  c_{a}\\left(\\frac{h}{k}\\right) \\ = \\\n  k^{a}\\sum_{m=1}^{k-1}\\cot\\left(\\frac{\\pi mh}{k}\\right)\\zeta\\left(-a,\\frac{m}{k}\\right), \\] where $a\\in\\mathbb{C}$, $h$ and $k$ are positive coprime integers, and $\\zeta(a,x)$ denotes the Hurwitz zeta function. We derive a new reciprocity theorem for these Bettin--Conrey sums, which in the case of an odd negative integer $a$ can be explicitly given in terms of Bernoulli numbers. This, in turn, implies explicit formulas for the peri"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1601.06839","kind":"arxiv","version":3},"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"}