{"paper":{"title":"Practical Computations of the Mertens Function: $M(10^{24})$ and $M(10^{25})$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Greg Hurst","submitted_at":"2026-07-08T15:57:01Z","abstract_excerpt":"The Mertens function is defined as $M(x)=\\sum_{n\\leq x}\\mu(n)$, where $\\mu(n)$ is the M\\\"obius function. This paper describes a practical implementation of the classical $O(x^{2/3+\\varepsilon})$ algorithm for computing $M(x)$ at isolated values, together with the segmented M\\\"obius and Mertens sieve on which it relies. The implementation was used to compute $M(10^{24}) = 7189337839$ and $M(10^{25}) = -258560632948$, taking $7.0$ days and $34.6$ days, respectively. These computations extend the previous record of $M(10^{23})$ by two orders of magnitude. Run standalone, the segmented sieve compu"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.07566","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/2607.07566/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"}