{"paper":{"title":"Moments of the derivative of the characteristic polynomial of unitary matrices","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MP","math.NT"],"primary_cat":"math-ph","authors_text":"Brian Conrey, Emilia Alvarez, Michael O. Rubinstein, Nina C. Snaith","submitted_at":"2024-07-18T03:25:55Z","abstract_excerpt":"Let $\\Lambda_X(s)=\\det(I-sX^{\\dagger})$ be the characteristic polynomial of a Haar distributed unitary matrix $X$. It is believed that the distribution of values of $\\Lambda_X(s)$ model the distribution of values of the Riemann zeta-function $\\zeta(s)$. This principle motivates many avenues of study. Of particular interest is the behavior of $\\Lambda_X'(s)$ and the distribution of its zeros (all of which lie inside or on the unit circle). In this article we present several identities for the moments of $\\Lambda_X'(s)$ averaged over $U(N)$, for $s \\in \\mathbb{C}$ as well as specialized to $|s|="},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2407.13124","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2407.13124/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"}