{"paper":{"title":"Murmurations of Mestre-Nagao sums","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Lukas Novak, Matija Kazalicki, Zvonimir Bujanovi\\'c","submitted_at":"2024-03-26T12:03:50Z","abstract_excerpt":"This paper investigates the detection of the rank of elliptic curves with ranks 0 and 1, employing a heuristic known as the Mestre-Nagao sum\n  \\[ S(B) = \\frac{1}{\\log{B}} \\sum_{\\substack{p<B \\\\ \\textrm{good reduction}}} \\frac{a_p(E)\\log{p}}{p}, \\]\nwhere $a_p(E)$ is defined as $p + 1 - \\#E(\\mathbb{F}_p)$ for an elliptic curve $E/\\mathbb{Q}$ with good reduction at prime $p$. This approach is inspired by the Birch and Swinnerton-Dyer conjecture.\n  Our observations reveal an oscillatory behavior in the sums, closely associated with the recently discovered phenomena of murmurations of elliptic curv"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2403.17626","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2403.17626/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"}