{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,22]],"date-time":"2026-04-22T08:47:56Z","timestamp":1776847676846,"version":"3.51.2"},"reference-count":14,"publisher":"American Mathematical Society (AMS)","issue":"272","license":[{"start":{"date-parts":[[2011,4,14]],"date-time":"2011-04-14T00:00:00Z","timestamp":1302739200000},"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                    In this article, we study the problem of changes 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: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: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                    . We provide three improvements. First, we give better esimates of error term for Lehman\u2019s theorem. Second, we rigorously prove the positivity of this difference for a region formerly conjectured by Patrick Demichel. Third, we improve the estimates for regions of positivity by using number theoretic results.\n                  <\/p>","DOI":"10.1090\/s0025-5718-10-02351-3","type":"journal-article","created":{"date-parts":[[2010,7,6]],"date-time":"2010-07-06T17:17:12Z","timestamp":1278436632000},"page":"2395-2405","source":"Crossref","is-referenced-by-count":13,"title":["A sharp region where \ud835\udf0b(\ud835\udc65)-fc(r) is positive"],"prefix":"10.1090","volume":"79","author":[{"given":"Yannick","family":"Saouter","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":[[2010,4,14]]},"reference":[{"issue":"1","key":"1","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":"2","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"},{"issue":"177","key":"3","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"},{"issue":"231","key":"4","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. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"key":"5","unstructured":"K.F. Chao and R. Plymen. A new bound for the smallest \ud835\udc65 with \ud835\udf0b(\ud835\udc65)>fc(r). arXiv:math\/0509312 [math.NT], Submitted, 2005."},{"key":"6","unstructured":"P. Demichel. The prime counting function and related subjects. Available at http:\/\/www. mybloop.com\/dmlpat\/maths\/li_crossover_pi.pdf, 2005."},{"key":"7","first-page":"64","article-title":"Approximate formulas for some functions of prime numbers","volume":"6","author":"Rosser, J. Barkley","year":"1962","journal-title":"Illinois J. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0019-2082","issn-type":"print"},{"issue":"4","key":"8","doi-asserted-by":"publisher","first-page":"373","DOI":"10.4064\/aa-94-4-373-381","article-title":"Inequalities concerning the function \ud835\udf0b(\ud835\udc65): applications","volume":"94","author":"Panaitopol, Lauren\u0163iu","year":"2000","journal-title":"Acta Arith.","ISSN":"https:\/\/id.crossref.org\/issn\/0065-1036","issn-type":"print"},{"key":"9","unstructured":"P. Dusart. Autour de la fonction qui compte le nombre de nombres premiers. Ph.D. thesis, Universit\u00e9 de Limoges, 1998."},{"key":"10","unstructured":"J. van de Lune. Unpublished, 2001."},{"key":"11","unstructured":"X. Gourdon and P. Demichel. The first 10\u00b9\u00b3 zeros of the Riemann Zeta function, and zeros computation at very large height. Available at http:\/\/numbers.computation.free.fr\/ Constants\/Miscellaneous\/zetazeros1e131e 24.pdf, 2004."},{"key":"12","unstructured":"S. Wedeniwski. Zetagrid home page. http:\/\/www.zetagrid.net\/, 2005."},{"issue":"1-2","key":"13","doi-asserted-by":"publisher","first-page":"247","DOI":"10.1016\/S0377-0427(00)00336-8","article-title":"Computational strategies for the Riemann zeta function","volume":"121","author":"Borwein, Jonathan M.","year":"2000","journal-title":"J. Comput. Appl. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0377-0427","issn-type":"print"},{"issue":"134","key":"14","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. 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