{"paper":{"title":"Computations of the Mertens Function and Improved Bounds on the Mertens Conjecture","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Greg Hurst","submitted_at":"2016-10-26T21:14:02Z","abstract_excerpt":"The Mertens function is defined as $M(x) = \\sum_{n \\leq x} \\mu(n)$, where $\\mu(n)$ is the M\\\"obius function. The Mertens conjecture states $|M(x)/\\sqrt{x}| < 1$ for $x > 1$, which was proven false in 1985 by showing $\\liminf M(x)/\\sqrt{x} < -1.009$ and $\\limsup M(x)/\\sqrt{x} > 1.06$. The same techniques used were revisited here with present day hardware and algorithms, giving improved lower and upper bounds of $-1.837625$ and $1.826054$. In addition, $M(x)$ was computed for all $x \\leq 10^{16}$, recording all extrema, all zeros, and $10^8$ values sampled at a regular interval. Lastly, an algor"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1610.08551","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":""},"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"}