{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,22]],"date-time":"2026-04-22T08:20:14Z","timestamp":1776846014122,"version":"3.51.2"},"reference-count":12,"publisher":"American Mathematical Society (AMS)","issue":"295","license":[{"start":{"date-parts":[[2016,2,12]],"date-time":"2016-02-12T00:00:00Z","timestamp":1455235200000},"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":"<p>\n                    We consider the least number\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"x\">\n                        <mml:semantics>\n                          <mml:mi>x<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">x<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    for which a change of sign of\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"pi left-parenthesis x right-parenthesis minus normal l normal i left-parenthesis x right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>\n                              \u03c0\n                              \n                            <\/mml:mi>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>x<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                            <mml:mo>\n                              \u2212\n                              \n                            <\/mml:mo>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mi mathvariant=\"normal\">l<\/mml:mi>\n                              <mml:mi mathvariant=\"normal\">i<\/mml:mi>\n                            <\/mml:mrow>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>x<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\pi (x)-\\mathrm {li}(x)<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    occurs. First, we consider modifications of Lehman\u2019s method that enable us to obtain better estimates of some error terms. Second, we establish a new smaller upper bound for the first\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"x\">\n                        <mml:semantics>\n                          <mml:mi>x<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">x<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    for which the difference is positive. Third, we use numerical computations to improve the final result.\n                  <\/p>","DOI":"10.1090\/s0025-5718-2015-02930-5","type":"journal-article","created":{"date-parts":[[2015,2,12]],"date-time":"2015-02-12T14:17:08Z","timestamp":1423750628000},"page":"2433-2446","source":"Crossref","is-referenced-by-count":15,"title":["A still sharper region where \ud835\udf0b(\ud835\udc65)-fc(r) is positive"],"prefix":"10.1090","volume":"84","author":[{"given":"Yannick","family":"Saouter","sequence":"first","affiliation":[]},{"given":"Timothy","family":"Trudgian","sequence":"additional","affiliation":[]},{"given":"Patrick","family":"Demichel","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2015,2,12]]},"reference":[{"issue":"1","key":"1","doi-asserted-by":"publisher","first-page":"345","DOI":"10.1007\/BF02422950","article-title":"\u00dcber die Nullstellen der Riemannschen Zetafunktion","volume":"41","author":"Backlund, R. J.","year":"1916","journal-title":"Acta Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0001-5962","issn-type":"print"},{"key":"2","unstructured":"[{Dus}98] P. Dusart, Autour de la fonction qui compte le nombre de nombres premiers, Ph.D. thesis, Universit\u00e9 de Limoges, 1998."},{"key":"3","unstructured":"[GD04] X. Gourdon and P. Demichel, The first 10\u00b9\u00b3 zeros of the Riemann Zeta function, and zeros computation at very large height. http:\/\/numbers.computation.free.fr\/Constants\/Miscellaneous\/zetazeros1e131e24.pdf, 2004."},{"issue":"1","key":"4","doi-asserted-by":"publisher","first-page":"55","DOI":"10.1007\/s10444-007-9039-2","article-title":"The prime-counting function and its analytic approximations: \ud835\udf0b(\ud835\udc65) and its approximations","volume":"29","author":"Kotnik, Tadej","year":"2008","journal-title":"Adv. Comput. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/1019-7168","issn-type":"print"},{"key":"5","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":"6","unstructured":"[{Lit}14] J. E. Littlewood, Sur la distribution des nombres premiers, Comptes Rendus 158 (1914), 1869\u20131872."},{"key":"7","unstructured":"[Map] Maple, Maplesoft, a division of Waterloo Maple Inc., Waterloo, Ontario."},{"key":"8","unstructured":"[{Pla}12] D. J. Platt, Computing \ud835\udf0b(\ud835\udc65) analytically, http:\/\/arxiv.org\/abs\/1203.5712. See also http\/\/www.lmfdb.org\/zeros\/zeta, 2012."},{"issue":"134","key":"9","doi-asserted-by":"publisher","first-page":"337","DOI":"10.2307\/2005976","article-title":"Sharper bounds for the Chebyshev functions \ud835\udf03(\ud835\udc65) and \ud835\udf13(\ud835\udc65). II","volume":"30","author":"Schoenfeld, Lowell","year":"1976","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"272","key":"10","doi-asserted-by":"publisher","first-page":"2395","DOI":"10.1090\/S0025-5718-10-02351-3","article-title":"A sharp region where \ud835\udf0b(\ud835\udc65)-\ud835\udc59\ud835\udc56(\ud835\udc65) is positive","volume":"79","author":"Saouter, Yannick","year":"2010","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"278","key":"11","doi-asserted-by":"publisher","first-page":"1053","DOI":"10.1090\/S0025-5718-2011-02537-8","article-title":"An improved upper bound for the argument of the Riemann zeta-function on the critical line","volume":"81","author":"Trudgian, Timothy","year":"2012","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"key":"12","unstructured":"[vdL01] J. van de Lune, Unpublished, 2001."}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/www.ams.org\/mcom\/2015-84-295\/S0025-5718-2015-02930-5\/S0025-5718-2015-02930-5.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/mcom\/2015-84-295\/S0025-5718-2015-02930-5\/S0025-5718-2015-02930-5.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T18:28:57Z","timestamp":1776796137000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/2015-84-295\/S0025-5718-2015-02930-5\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2015,2,12]]},"references-count":12,"journal-issue":{"issue":"295","published-print":{"date-parts":[[2015,9]]}},"alternative-id":["S0025-5718-2015-02930-5"],"URL":"https:\/\/doi.org\/10.1090\/s0025-5718-2015-02930-5","archive":["CLOCKSS","Portico"],"relation":{},"ISSN":["1088-6842","0025-5718"],"issn-type":[{"value":"1088-6842","type":"electronic"},{"value":"0025-5718","type":"print"}],"subject":[],"published":{"date-parts":[[2015,2,12]]}}}