{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,22]],"date-time":"2026-04-22T08:12:22Z","timestamp":1776845542879,"version":"3.51.2"},"reference-count":10,"publisher":"American Mathematical Society (AMS)","issue":"225","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>\n                    <sc>Rosser<\/sc>\n                    and\n                    <sc>Schoenfeld<\/sc>\n                    have used the fact that the first 3,500,000 zeros of the\n                    <sc>Riemann<\/sc>\n                    zeta function lie on the critical line to give estimates on\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"psi left-parenthesis x right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>\n                              \u03c8\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:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\psi (x)<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    and\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"theta left-parenthesis x right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>\n                              \u03b8\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:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\theta (x)<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    . With an improvement of the above result by\n                    <sc>Brent<\/sc>\n                    <italic>et al.<\/italic>\n                    , we are able to improve these estimates and to show that the\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"k Superscript t h\">\n                        <mml:semantics>\n                          <mml:msup>\n                            <mml:mi>k<\/mml:mi>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mi>t<\/mml:mi>\n                              <mml:mi>h<\/mml:mi>\n                            <\/mml:mrow>\n                          <\/mml:msup>\n                          <mml:annotation encoding=\"application\/x-tex\">k^{th}<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    prime is greater than\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"k left-parenthesis ln k plus ln ln k minus 1 right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>k<\/mml:mi>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>ln<\/mml:mi>\n                            <mml:mo>\n                              \u2061\n                              \n                            <\/mml:mo>\n                            <mml:mi>k<\/mml:mi>\n                            <mml:mo>+<\/mml:mo>\n                            <mml:mi>ln<\/mml:mi>\n                            <mml:mo>\n                              \u2061\n                              \n                            <\/mml:mo>\n                            <mml:mi>ln<\/mml:mi>\n                            <mml:mo>\n                              \u2061\n                              \n                            <\/mml:mo>\n                            <mml:mi>k<\/mml:mi>\n                            <mml:mo>\n                              \u2212\n                              \n                            <\/mml:mo>\n                            <mml:mn>1<\/mml:mn>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">k(\\ln k +\\ln \\ln k -1)<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    for\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"k greater-than-or-equal-to 2\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>k<\/mml:mi>\n                            <mml:mo>\n                              \u2265\n                              \n                            <\/mml:mo>\n                            <mml:mn>2<\/mml:mn>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">k\\geq 2<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    . We give further results without proof.\n                  <\/p>","DOI":"10.1090\/s0025-5718-99-01037-6","type":"journal-article","created":{"date-parts":[[2002,7,26]],"date-time":"2002-07-26T18:14:44Z","timestamp":1027707284000},"page":"411-415","source":"Crossref","is-referenced-by-count":98,"title":["The \ud835\udc58^{\ud835\udc61\u210e} prime is greater than \ud835\udc58(ln\ud835\udc58+lnln\ud835\udc58-1) for \ud835\udc58\u22652"],"prefix":"10.1090","volume":"68","author":[{"given":"Pierre","family":"Dusart","sequence":"first","affiliation":[]}],"member":"14","published-online":{"date-parts":[[1999]]},"reference":[{"issue":"160","key":"1","doi-asserted-by":"publisher","first-page":"681","DOI":"10.2307\/2007345","article-title":"On the zeros of the Riemann zeta function in the critical strip. II","volume":"39","author":"Brent, R. P.","year":"1982","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"key":"2","unstructured":"M. Cipolla, La determinazione assintotica dell\u2019\ud835\udc5b^{\ud835\udc56\ud835\udc5a\ud835\udc5c} numero primo, Matematiche Napoli 3 (1902), 132-166."},{"issue":"174","key":"3","doi-asserted-by":"publisher","first-page":"667","DOI":"10.2307\/2008005","article-title":"On the zeros of the Riemann zeta function in the critical strip. IV","volume":"46","author":"van de Lune, J.","year":"1986","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"1","key":"4","doi-asserted-by":"publisher","first-page":"215","DOI":"10.5802\/jtnb.166","article-title":"Bornes effectives pour certaines fonctions concernant les nombres premiers","volume":"8","author":"Massias, Jean-Pierre","year":"1996","journal-title":"J. Th\\'{e}or. Nombres Bordeaux","ISSN":"https:\/\/id.crossref.org\/issn\/1246-7405","issn-type":"print"},{"issue":"4","key":"5","doi-asserted-by":"publisher","first-page":"367","DOI":"10.4064\/aa-42-4-367-389","article-title":"Estimation de la fonction de Tchebychef \ud835\udf03 sur le \ud835\udc58-i\u00e8me nombre premier et grandes valeurs de la fonction \ud835\udf14(\ud835\udc5b) nombre de diviseurs premiers de \ud835\udc5b","volume":"42","author":"Robin, Guy","year":"1983","journal-title":"Acta Arith.","ISSN":"https:\/\/id.crossref.org\/issn\/0065-1036","issn-type":"print"},{"key":"6","doi-asserted-by":"crossref","unstructured":"J. B. Rosser, The \ud835\udc5b-th prime is greater than \ud835\udc5blog\ud835\udc5b, Proc. London Math. Soc. (2) 45 (1939), 21-44.","DOI":"10.1112\/plms\/s2-45.1.21"},{"key":"7","volume-title":"New First Course in the Theory of Equations","author":"Dickson, Leonard Eugene","year":"1939"},{"key":"8","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"},{"key":"9","doi-asserted-by":"publisher","first-page":"426","DOI":"10.1007\/BF01695512","article-title":"\u00dcber eine Minimalaufgabe im Gebiete der analytischen Funktionen von mehreren Ver\u00e4nderlichen","volume":"47","author":"Wirtinger, Wilhelm","year":"1939","journal-title":"Monatsh. Math. Phys.","ISSN":"https:\/\/id.crossref.org\/issn\/1812-8076","issn-type":"print"},{"key":"10","doi-asserted-by":"publisher","first-page":"426","DOI":"10.1007\/BF01695512","article-title":"\u00dcber eine Minimalaufgabe im Gebiete der analytischen Funktionen von mehreren Ver\u00e4nderlichen","volume":"47","author":"Wirtinger, Wilhelm","year":"1939","journal-title":"Monatsh. Math. Phys.","ISSN":"https:\/\/id.crossref.org\/issn\/1812-8076","issn-type":"print"}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/www.ams.org\/mcom\/1999-68-225\/S0025-5718-99-01037-6\/S0025-5718-99-01037-6.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/mcom\/1999-68-225\/S0025-5718-99-01037-6\/S0025-5718-99-01037-6.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,20]],"date-time":"2026-04-20T21:58:27Z","timestamp":1776722307000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/1999-68-225\/S0025-5718-99-01037-6\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1999]]},"references-count":10,"journal-issue":{"issue":"225","published-print":{"date-parts":[[1999,1]]}},"alternative-id":["S0025-5718-99-01037-6"],"URL":"https:\/\/doi.org\/10.1090\/s0025-5718-99-01037-6","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":[[1999]]}}}