{"paper":{"title":"Computing $\\pi(N)$: An elementary approach in $\\tilde{O}(\\sqrt{N})$ time","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Dean Hirsch, Ido Kessler, Uri Mendlovic","submitted_at":"2022-12-19T21:05:14Z","abstract_excerpt":"We present an efficient and elementary algorithm for computing the number of primes up to $N$ in $\\tilde{O}(\\sqrt N)$ time, improving upon the existing combinatorial methods that require $\\tilde{O}(N ^ {2/3})$ time. Our method has a similar time complexity to the analytical approach to prime counting, while avoiding complex analysis and the use of arbitrary precision complex numbers. While the most time-efficient version of our algorithm requires $\\tilde{O}(\\sqrt N)$ space, we present a continuous space-time trade-off, showing, e.g., how to reduce the space complexity to $\\tilde{O}(\\sqrt[3]{N}"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2212.09857","kind":"arxiv","version":4},"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/2212.09857/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"}