{"paper":{"title":"Correlations of the Riemann zeta function","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Michael J. Curran","submitted_at":"2023-03-17T17:03:50Z","abstract_excerpt":"Assuming the Riemann hypothesis, we investigate the shifted moments of the zeta function \\[ M_{\\alpha,{\\beta}}(T) = \\int_T^{2T} \\prod_{k = 1}^m |\\zeta(\\tfrac{1}{2} + i (t + \\alpha_k))|^{2 \\beta_k} dt \\] introduced by Chandee, where ${\\alpha} = {\\alpha}(T) = (\\alpha_1, \\ldots, \\alpha_m)$ and ${\\beta} = (\\beta_1 \\ldots , \\beta_m)$ satisfy $|\\alpha_k| \\leq T/2$ and $\\beta_k\\geq 0$. We shall prove that \\[ M_{{\\alpha},{\\beta}}(T) \\ll_{{\\beta}} T (\\log T)^{\\beta_1^2 + \\cdots + \\beta_m^2} \\prod_{1\\leq j < k \\leq m} |\\zeta(1 + i(\\alpha_j - \\alpha_k) + 1/ \\log T )|^{2\\beta_j \\beta_k}. \\] This improves "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2303.10123","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/2303.10123/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"}