{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,17]],"date-time":"2026-08-17T23:11:22Z","timestamp":1787008282319,"version":"build-2736575974"},"reference-count":17,"publisher":"American Mathematical Society (AMS)","issue":"310","license":[{"start":{"date-parts":[[2018,11,1]],"date-time":"2018-11-01T00:00:00Z","timestamp":1541030400000},"content-version":"am","delay-in-days":365,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"

\n The Mertens function is defined as\n \n \n \n \n M<\/mml:mi>\n (<\/mml:mo>\n x<\/mml:mi>\n )<\/mml:mo>\n =<\/mml:mo>\n \n \n \u2211\n \n <\/mml:mo>\n \n n<\/mml:mi>\n \n \u2264\n \n <\/mml:mo>\n x<\/mml:mi>\n <\/mml:mrow>\n <\/mml:munder>\n \n \u03bc\n \n <\/mml:mi>\n (<\/mml:mo>\n n<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n M(x) = \\sum _{n \\leq x} \\mu (n)<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n , where\n \n \n \n \n \n \u03bc\n \n <\/mml:mi>\n (<\/mml:mo>\n n<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n \\mu (n)<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n is the M\u00f6bius function. The Mertens conjecture states\n \n \n \n \n \n |<\/mml:mo>\n <\/mml:mrow>\n M<\/mml:mi>\n (<\/mml:mo>\n x<\/mml:mi>\n )<\/mml:mo>\n \n \/<\/mml:mo>\n <\/mml:mrow>\n \n x<\/mml:mi>\n <\/mml:msqrt>\n \n |<\/mml:mo>\n <\/mml:mrow>\n ><\/mml:mo>\n 1<\/mml:mn>\n <\/mml:mrow>\n |M(x)\/\\sqrt {x}| > 1<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n for\n \n \n \n \n x<\/mml:mi>\n ><\/mml:mo>\n 1<\/mml:mn>\n <\/mml:mrow>\n x > 1<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n , which was proven false in 1985 by showing\n \n \n \n \n lim\u2006inf<\/mml:mo>\n M<\/mml:mi>\n (<\/mml:mo>\n x<\/mml:mi>\n )<\/mml:mo>\n \n \/<\/mml:mo>\n <\/mml:mrow>\n \n x<\/mml:mi>\n <\/mml:msqrt>\n ><\/mml:mo>\n \n \u2212\n \n <\/mml:mo>\n 1.009<\/mml:mn>\n <\/mml:mrow>\n \\liminf M(x)\/\\sqrt {x} > -1.009<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n and\n \n \n \n \n lim\u2006sup<\/mml:mo>\n M<\/mml:mi>\n (<\/mml:mo>\n x<\/mml:mi>\n )<\/mml:mo>\n \n \/<\/mml:mo>\n <\/mml:mrow>\n \n x<\/mml:mi>\n <\/mml:msqrt>\n ><\/mml:mo>\n 1.06<\/mml:mn>\n <\/mml:mrow>\n \\limsup M(x)\/\\sqrt {x} > 1.06<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n . The same techniques used were revisited here with present day hardware and algorithms, giving improved lower and upper bounds of\n \n \n \n \n \n \u2212\n \n <\/mml:mo>\n 1.837625<\/mml:mn>\n <\/mml:mrow>\n -1.837625<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n and\n \n \n \n 1.826054<\/mml:mn>\n 1.826054<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n . In addition,\n \n \n \n \n M<\/mml:mi>\n (<\/mml:mo>\n x<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n M(x)<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n was computed for all\n \n \n \n \n x<\/mml:mi>\n \n \u2264\n \n <\/mml:mo>\n \n 10<\/mml:mn>\n \n 16<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msup>\n <\/mml:mrow>\n x \\leq 10^{16}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n , recording all extrema, all zeros, and\n \n \n \n \n 10<\/mml:mn>\n 8<\/mml:mn>\n <\/mml:msup>\n 10^8<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n values sampled at a regular interval. Finally, an algorithm to compute\n \n \n \n \n M<\/mml:mi>\n (<\/mml:mo>\n x<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n M(x)<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n in\n \n \n \n \n O<\/mml:mi>\n (<\/mml:mo>\n \n x<\/mml:mi>\n \n 2<\/mml:mn>\n \n \/<\/mml:mo>\n <\/mml:mrow>\n 3<\/mml:mn>\n +<\/mml:mo>\n \n \u03b5\n \n <\/mml:mi>\n <\/mml:mrow>\n <\/mml:msup>\n )<\/mml:mo>\n <\/mml:mrow>\n O(x^{2\/3+\\varepsilon })<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n time was used on all powers of two up to\n \n \n \n \n 2<\/mml:mn>\n \n 73<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msup>\n 2^{73}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n .\n <\/p>","DOI":"10.1090\/mcom\/3275","type":"journal-article","created":{"date-parts":[[2017,4,5]],"date-time":"2017-04-05T11:41:22Z","timestamp":1491392482000},"page":"1013-1028","source":"Crossref","is-referenced-by-count":17,"title":["Computations of the Mertens function and improved bounds on the Mertens conjecture"],"prefix":"10.1090","volume":"87","author":[{"given":"Greg","family":"Hurst","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"14","published-online":{"date-parts":[[2017,11,1]]},"reference":[{"key":"1","unstructured":"E. 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Montgomery, Division by Invariant Integers using Multiplication Proceedings of SIGPLAN \u201894 Conference on Programming Language Design and Implementation."},{"key":"9","unstructured":"Richard Sladkey, A Successive Approximation Algorithm for Computing the Divisor Summatory Function (arXiv:1206.3369 [math.NT], Jun 2012) 13\u201316"},{"key":"10","unstructured":"The FPLLL development team, fplll, a lattice reduction library (available at \\url{https:\/\/github.com\/fplll\/fplll}) (January 2016)"},{"key":"11","isbn-type":"print","doi-asserted-by":"publisher","first-page":"271","DOI":"10.1145\/1576702.1576740","article-title":"H-LLL: using Householder inside LLL","author":"Morel, Ivan","year":"2009","ISBN":"https:\/\/id.crossref.org\/isbn\/9781605586090"},{"key":"12","doi-asserted-by":"publisher","first-page":"1","DOI":"10.2307\/1994409","article-title":"Oscillation theorems of arithmetical functions","volume":"126","author":"Grosswald, Emil","year":"1967","journal-title":"Trans. Amer. 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