{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,22]],"date-time":"2026-04-22T08:36:59Z","timestamp":1776847019471,"version":"3.51.2"},"reference-count":32,"publisher":"American Mathematical Society (AMS)","issue":"276","license":[{"start":{"date-parts":[[2012,4,1]],"date-time":"2012-04-01T00:00:00Z","timestamp":1333238400000},"content-version":"am","delay-in-days":366,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"

\n An analysis of the local variations of the prime counting function\n \n \n \n \n \n \u03c0\n \n <\/mml:mi>\n (<\/mml:mo>\n x<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n \\pi (x)<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n due to the impact of the non-trivial, complex zeros\n \n \n \n \n \n \u03f1\n \n <\/mml:mi>\n k<\/mml:mi>\n <\/mml:msub>\n \\varrho _k<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n of\n \n \n \n \n \n \u03b6\n \n <\/mml:mi>\n (<\/mml:mo>\n s<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n \\zeta (s)<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n is provided for\n \n \n \n \n x<\/mml:mi>\n ><\/mml:mo>\n \n 10<\/mml:mn>\n \n \n 10<\/mml:mn>\n \n 13<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msup>\n <\/mml:mrow>\n <\/mml:msup>\n <\/mml:mrow>\n x>10^{10^{13}}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n using up to 200 billion\n \n \n \n \n \n \u03b6\n \n <\/mml:mi>\n (<\/mml:mo>\n s<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n \\zeta (s)<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n complex zeros. A new bound for\n \n \n \n \n \n |<\/mml:mo>\n <\/mml:mrow>\n \n l<\/mml:mi>\n i<\/mml:mi>\n <\/mml:mrow>\n (<\/mml:mo>\n x<\/mml:mi>\n )<\/mml:mo>\n \n \u2212\n \n <\/mml:mo>\n \n \u03c0\n \n <\/mml:mi>\n (<\/mml:mo>\n x<\/mml:mi>\n )<\/mml:mo>\n \n |<\/mml:mo>\n <\/mml:mrow>\n ><\/mml:mo>\n \n x<\/mml:mi>\n \n 1<\/mml:mn>\n \n \/<\/mml:mo>\n <\/mml:mrow>\n 2<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msup>\n (<\/mml:mo>\n \n l<\/mml:mi>\n o<\/mml:mi>\n g<\/mml:mi>\n <\/mml:mrow>\n \n \n l<\/mml:mi>\n o<\/mml:mi>\n g<\/mml:mi>\n <\/mml:mrow>\n \n \n l<\/mml:mi>\n o<\/mml:mi>\n g<\/mml:mi>\n <\/mml:mrow>\n \n x<\/mml:mi>\n +<\/mml:mo>\n e<\/mml:mi>\n +<\/mml:mo>\n 1<\/mml:mn>\n )<\/mml:mo>\n \n \/<\/mml:mo>\n <\/mml:mrow>\n e<\/mml:mi>\n \n \n l<\/mml:mi>\n o<\/mml:mi>\n g<\/mml:mi>\n <\/mml:mrow>\n \n x<\/mml:mi>\n <\/mml:mrow>\n |\\mathrm {li}(x)-\\pi (x)|>x^{1\/2}(\\mathrm {log}\\,\\mathrm {log}\\,\\mathrm {log}\\,x+e+1)\/e\\, \\mathrm {log}\\,x<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n is proposed consistent with the error growth rate in Littlewood\u2019s proof that\n \n \n \n \n \n l<\/mml:mi>\n i<\/mml:mi>\n <\/mml:mrow>\n (<\/mml:mo>\n x<\/mml:mi>\n )<\/mml:mo>\n \n \u2212\n \n <\/mml:mo>\n \n \u03c0\n \n <\/mml:mi>\n (<\/mml:mo>\n x<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n \\mathrm {li}(x)-\\pi (x)<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n changes sign infinitely often. This bound is also consistent with all presently known cases where\n \n \n \n \n \n \u03c0\n \n <\/mml:mi>\n (<\/mml:mo>\n x<\/mml:mi>\n )<\/mml:mo>\n ><\/mml:mo>\n \n l<\/mml:mi>\n i<\/mml:mi>\n <\/mml:mrow>\n (<\/mml:mo>\n x<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n \\pi (x)>\\mathrm {li}(x)<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n including many new examples listed. This implies that Littlewood\u2019s constant\n \n \n \n \n \n K<\/mml:mi>\n <\/mml:mrow>\n =<\/mml:mo>\n 1<\/mml:mn>\n \n \/<\/mml:mo>\n <\/mml:mrow>\n e<\/mml:mi>\n <\/mml:mrow>\n \\mathrm {K}=1\/e<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n , the lower bound for Skewes\u2019 number is\n \n \n \n \n 3.17<\/mml:mn>\n \n \u00d7\n \n <\/mml:mo>\n \n 10<\/mml:mn>\n \n 114<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msup>\n <\/mml:mrow>\n 3.17\\times 10^{114}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n and the positive constant\n \n \n \n c<\/mml:mi>\n c<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n in the Riemann Hypothesis equivalent\n \n \n \n \n \n |<\/mml:mo>\n <\/mml:mrow>\n \n l<\/mml:mi>\n i<\/mml:mi>\n <\/mml:mrow>\n (<\/mml:mo>\n x<\/mml:mi>\n )<\/mml:mo>\n \n \u2212\n \n <\/mml:mo>\n \n \u03c0\n \n <\/mml:mi>\n (<\/mml:mo>\n x<\/mml:mi>\n )<\/mml:mo>\n \n |<\/mml:mo>\n <\/mml:mrow>\n ><\/mml:mo>\n c<\/mml:mi>\n \n \n l<\/mml:mi>\n o<\/mml:mi>\n g<\/mml:mi>\n <\/mml:mrow>\n (<\/mml:mo>\n x<\/mml:mi>\n )<\/mml:mo>\n \n x<\/mml:mi>\n \n 1<\/mml:mn>\n \n \/<\/mml:mo>\n <\/mml:mrow>\n 2<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msup>\n <\/mml:mrow>\n |\\mathrm {li}(x)-\\pi (x)|>c\\,\\mathrm {log}(x)x^{1\/2}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n is less than\n \n \n \n \n 3<\/mml:mn>\n \n \u00d7\n \n <\/mml:mo>\n \n 10<\/mml:mn>\n \n \n \u2212\n \n <\/mml:mo>\n 27<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msup>\n <\/mml:mrow>\n 3\\times 10^{-27}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n .\n <\/p>","DOI":"10.1090\/s0025-5718-2011-02477-4","type":"journal-article","created":{"date-parts":[[2011,4,1]],"date-time":"2011-04-01T08:21:15Z","timestamp":1301646075000},"page":"2381-2394","source":"Crossref","is-referenced-by-count":3,"title":["The impact of \ud835\udf01(\ud835\udc60) complex zeros on \ud835\udf0b(\ud835\udc65) for \ud835\udc65<10^{10\u00b9\u00b3}"],"prefix":"10.1090","volume":"80","author":[{"given":"Douglas","family":"Stoll","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Patrick","family":"Demichel","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"14","published-online":{"date-parts":[[2011,4,1]]},"reference":[{"issue":"231","key":"1","doi-asserted-by":"publisher","first-page":"1285","DOI":"10.1090\/S0025-5718-99-01104-7","article-title":"A new bound for the smallest \ud835\udc65 with \ud835\udf0b(\ud835\udc65)>\ud835\udc59\ud835\udc56(\ud835\udc65)","volume":"69","author":"Bays, Carter","year":"2000","journal-title":"Math. 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Kulsha, Unpublished paper and tables of minimum and maximum \ud835\udf0b(\ud835\udc65) variations, (http:\/\/www.primefan.ru\/stuff\/primes\/table.html) (updated 7\/4\/2010)."},{"key":"17","doi-asserted-by":"publisher","first-page":"397","DOI":"10.4064\/aa-11-4-397-410","article-title":"On the difference \ud835\udf0b(\ud835\udc65)-\ud835\udc59\ud835\udc56(\ud835\udc65)","volume":"11","author":"Lehman, R. Sherman","year":"1966","journal-title":"Acta Arith.","ISSN":"https:\/\/id.crossref.org\/issn\/0065-1036","issn-type":"print"},{"key":"18","unstructured":"J.E. Littlewood, Sur la distribution des nombres premiers, C. R. Acad. Sci. 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(3)","ISSN":"https:\/\/id.crossref.org\/issn\/0024-6115","issn-type":"print"},{"key":"30","unstructured":"T. O. e Silva, Tables of values of pi(x) and of pi\\textsubscript{2}(x), (http:\/\/\\allowbreak www.\\allowbreak ieeta.\\allowbreak pt\/\\allowbreak~tos\/\\allowbreak primes.\\allowbreak html), updated March 2010."},{"issue":"177","key":"31","doi-asserted-by":"publisher","first-page":"323","DOI":"10.2307\/2007893","article-title":"On the sign of the difference \ud835\udf0b(\ud835\udc65)-\ud835\udc59\ud835\udc56(\ud835\udc65)","volume":"48","author":"te Riele, Herman J. J.","year":"1987","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"key":"32","unstructured":"de la Vall\u00e9e Poussin, Recherches analytiques sur la th\u00e9orie des nombres premiers, Ann. Soc. Sci. 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