{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,22]],"date-time":"2026-04-22T15:08:28Z","timestamp":1776870508504,"version":"3.51.2"},"reference-count":47,"publisher":"American Mathematical Society (AMS)","issue":"331","license":[{"start":{"date-parts":[[2022,6,15]],"date-time":"2022-06-15T00:00:00Z","timestamp":1655251200000},"content-version":"am","delay-in-days":365,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"funder":[{"DOI":"10.13039\/501100000038","name":"Natural Sciences and Engineering Research Council of Canada","doi-asserted-by":"publisher","award":["RGPIN-2020-06731"],"award-info":[{"award-number":["RGPIN-2020-06731"]}],"id":[{"id":"10.13039\/501100000038","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100000038","name":"Natural Sciences and Engineering Research Council of Canada","doi-asserted-by":"publisher","award":["RGPIN-2020-06032"],"award-info":[{"award-number":["RGPIN-2020-06032"]}],"id":[{"id":"10.13039\/501100000038","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100000038","name":"Natural Sciences and Engineering Research Council of Canada","doi-asserted-by":"publisher","award":["ULRF G00003674"],"award-info":[{"award-number":["ULRF G00003674"]}],"id":[{"id":"10.13039\/501100000038","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100000038","name":"Natural Sciences and Engineering Research Council of Canada","doi-asserted-by":"publisher","award":["RGPIN-2020-06731"],"award-info":[{"award-number":["RGPIN-2020-06731"]}],"id":[{"id":"10.13039\/501100000038","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100000038","name":"Natural Sciences and Engineering Research Council of Canada","doi-asserted-by":"publisher","award":["RGPIN-2020-06032"],"award-info":[{"award-number":["RGPIN-2020-06032"]}],"id":[{"id":"10.13039\/501100000038","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100000038","name":"Natural Sciences and Engineering Research Council of Canada","doi-asserted-by":"publisher","award":["ULRF G00003674"],"award-info":[{"award-number":["ULRF G00003674"]}],"id":[{"id":"10.13039\/501100000038","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100008740","name":"University of Lethbridge","doi-asserted-by":"publisher","award":["RGPIN-2020-06731"],"award-info":[{"award-number":["RGPIN-2020-06731"]}],"id":[{"id":"10.13039\/100008740","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100008740","name":"University of Lethbridge","doi-asserted-by":"publisher","award":["RGPIN-2020-06032"],"award-info":[{"award-number":["RGPIN-2020-06032"]}],"id":[{"id":"10.13039\/100008740","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100008740","name":"University of Lethbridge","doi-asserted-by":"publisher","award":["ULRF G00003674"],"award-info":[{"award-number":["ULRF G00003674"]}],"id":[{"id":"10.13039\/100008740","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>\n                    In this article, we provide explicit bounds for the prime counting functions\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                    for all ranges of\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                    . The bounds for the error term for\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 minus x\">\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:mo>\n                              \u2212\n                              \n                            <\/mml:mo>\n                            <mml:mi>x<\/mml:mi>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\theta (x)- x<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    are of the shape\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"epsilon x\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>\n                              \u03b5\n                              \n                            <\/mml:mi>\n                            <mml:mi>x<\/mml:mi>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\varepsilon 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=\"StartFraction c Subscript k Baseline x Over left-parenthesis log x right-parenthesis Superscript k Baseline EndFraction\">\n                        <mml:semantics>\n                          <mml:mfrac>\n                            <mml:mrow>\n                              <mml:msub>\n                                <mml:mi>c<\/mml:mi>\n                                <mml:mi>k<\/mml:mi>\n                              <\/mml:msub>\n                              <mml:mi>x<\/mml:mi>\n                            <\/mml:mrow>\n                            <mml:mrow>\n                              <mml:mo stretchy=\"false\">(<\/mml:mo>\n                              <mml:mi>log<\/mml:mi>\n                              <mml:mo>\n                                \u2061\n                                \n                              <\/mml:mo>\n                              <mml:mi>x<\/mml:mi>\n                              <mml:msup>\n                                <mml:mo stretchy=\"false\">)<\/mml:mo>\n                                <mml:mi>k<\/mml:mi>\n                              <\/mml:msup>\n                            <\/mml:mrow>\n                          <\/mml:mfrac>\n                          <mml:annotation encoding=\"application\/x-tex\">\\frac {c_k x}{(\\log x)^k}<\/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 equals 1 comma ellipsis comma 5\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>k<\/mml:mi>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:mn>1<\/mml:mn>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mo>\n                              \u2026\n                              \n                            <\/mml:mo>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mn>5<\/mml:mn>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">k=1,\\ldots ,5<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    . Tables of values for\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"epsilon\">\n                        <mml:semantics>\n                          <mml:mi>\n                            \u03b5\n                            \n                          <\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">\\varepsilon<\/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=\"c Subscript k\">\n                        <mml:semantics>\n                          <mml:msub>\n                            <mml:mi>c<\/mml:mi>\n                            <mml:mi>k<\/mml:mi>\n                          <\/mml:msub>\n                          <mml:annotation encoding=\"application\/x-tex\">c_k<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    are provided.\n                  <\/p>","DOI":"10.1090\/mcom\/3643","type":"journal-article","created":{"date-parts":[[2021,3,24]],"date-time":"2021-03-24T11:25:49Z","timestamp":1616585149000},"page":"2281-2315","source":"Crossref","is-referenced-by-count":34,"title":["Sharper bounds for the Chebyshev function \ud835\udf03(\ud835\udc65)"],"prefix":"10.1090","volume":"90","author":[{"given":"Samuel","family":"Broadbent","sequence":"first","affiliation":[]},{"given":"Habiba","family":"Kadiri","sequence":"additional","affiliation":[]},{"given":"Allysa","family":"Lumley","sequence":"additional","affiliation":[]},{"given":"Nathan","family":"Ng","sequence":"additional","affiliation":[]},{"given":"Kirsten","family":"Wilk","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2021,6,15]]},"reference":[{"key":"1","first-page":"Paper No. A52, 21","article-title":"New estimates for some functions defined over primes","volume":"18","author":"Axler, Christian","year":"2018","journal-title":"Integers"},{"key":"2","unstructured":"S. Broadbent, A. Fiori, H. Kadiri, A. Lumley, N. Ng, J. Swidinsky, and K. Wilk, Tables of values of the Chebyshev functions \ud835\udf03(\ud835\udc65) and \ud835\udf13(\ud835\udc65), 41 pages,  arXiv:2002.11068, 2020."},{"issue":"301","key":"3","doi-asserted-by":"publisher","first-page":"2483","DOI":"10.1090\/mcom\/3060","article-title":"Estimating \ud835\udf0b(\ud835\udc65) and related functions under partial RH assumptions","volume":"85","author":"B\u00fcthe, Jan","year":"2016","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"312","key":"4","doi-asserted-by":"publisher","first-page":"1991","DOI":"10.1090\/mcom\/3264","article-title":"An analytic method for bounding \ud835\udf13(\ud835\udc65)","volume":"87","author":"B\u00fcthe, Jan","year":"2018","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"key":"5","unstructured":"P. L. Chebyshev, M\u00e9moire sur les nombres premiers, J. Math. Pures Appl. 17 (1852), 366\u2013390."},{"issue":"169","key":"6","doi-asserted-by":"publisher","first-page":"211","DOI":"10.2307\/2007805","article-title":"Estimates for the Chebyshev function \ud835\udf13(\ud835\udc65)-\ud835\udf03(\ud835\udc65)","volume":"44","author":"Costa Pereira, N.","year":"1985","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"4","key":"7","doi-asserted-by":"publisher","first-page":"307","DOI":"10.4064\/aa-52-4-307-337","article-title":"Elementary estimates for the Chebyshev function \ud835\udf13(\ud835\udc65) and for the M\u00f6bius function \ud835\udc40(\ud835\udc65)","volume":"52","author":"Costa Pereira, N.","year":"1989","journal-title":"Acta Arith.","ISSN":"https:\/\/id.crossref.org\/issn\/0065-1036","issn-type":"print"},{"key":"8","series-title":"Graduate Texts in Mathematics","isbn-type":"print","volume-title":"Multiplicative number theory","volume":"74","author":"Davenport, Harold","year":"2000","ISBN":"https:\/\/id.crossref.org\/isbn\/0387950974","edition":"3"},{"issue":"295","key":"9","doi-asserted-by":"publisher","first-page":"2433","DOI":"10.1090\/S0025-5718-2015-02930-5","article-title":"A still sharper region where \ud835\udf0b(\ud835\udc65)-\ud835\udc59\ud835\udc56(\ud835\udc65) is positive","volume":"84","author":"Saouter, Yannick","year":"2015","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"key":"10","doi-asserted-by":"crossref","unstructured":"J. M. Deshouillers, Retour sur la methode de Chebyshev, Bull. Soc. Math. France 49\u201350 (1977), 41\u201345.","DOI":"10.24033\/msmf.212"},{"issue":"2","key":"11","doi-asserted-by":"publisher","first-page":"177","DOI":"10.7169\/facm\/2016.55.2.3","article-title":"An explicit result for primes between cubes","volume":"55","author":"Dudek, Adrian W.","year":"2016","journal-title":"Funct. Approx. Comment. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0208-6573","issn-type":"print"},{"issue":"2","key":"12","first-page":"53","article-title":"In\u00e9galit\u00e9s explicites pour \ud835\udf13(\ud835\udc4b), \ud835\udf03(\ud835\udc4b), \ud835\udf0b(\ud835\udc4b) et les nombres premiers","volume":"21","author":"Dusart, Pierre","year":"1999","journal-title":"C. R. Math. Acad. Sci. Soc. R. Can.","ISSN":"https:\/\/id.crossref.org\/issn\/0706-1994","issn-type":"print"},{"key":"13","unstructured":"P. Dusart, Estimates of some functions over primes without R.H.,  arXiv:1002.0442 (February 2, 2010)."},{"issue":"298","key":"14","doi-asserted-by":"publisher","first-page":"875","DOI":"10.1090\/mcom\/3005","article-title":"Estimates of \ud835\udf13,\ud835\udf03 for large values of \ud835\udc65 without the Riemann hypothesis","volume":"85","author":"Dusart, Pierre","year":"2016","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"1","key":"15","doi-asserted-by":"publisher","first-page":"227","DOI":"10.1007\/s11139-016-9839-4","article-title":"Explicit estimates of some functions over primes","volume":"45","author":"Dusart, Pierre","year":"2018","journal-title":"Ramanujan J.","ISSN":"https:\/\/id.crossref.org\/issn\/1382-4090","issn-type":"print"},{"key":"16","unstructured":"P. Erd\u00f6s, Beweis eines Satzes von Tschebyschef, Acta Litt. Sci. Szeged 5 (1932), 194\u2013198."},{"issue":"293","key":"17","doi-asserted-by":"publisher","first-page":"1339","DOI":"10.1090\/S0025-5718-2014-02886-X","article-title":"New bounds for \ud835\udf13(\ud835\udc65)","volume":"84","author":"Faber, Laura","year":"2015","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"311","key":"18","doi-asserted-by":"publisher","first-page":"1451","DOI":"10.1090\/mcom\/3340","article-title":"Corrigendum to New bounds for \ud835\udf13(\ud835\udc65) [ MR3315511]","volume":"87","author":"Faber, Laura","year":"2018","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"key":"19","unstructured":"X. Gourdon, The 10\u00b9\u00b3 first zeros of the Riemann zeta function, and zeros computation at very large height, \\url{http:\/\/numbers.computation.free.fr\/Constants\/Miscellaneous\/zetazeros1e13-1e24.pdf}, 2004."},{"key":"20","isbn-type":"print","first-page":"407","article-title":"Estimates for the product of the primes not exceeding \ud835\udc65","author":"Grimson, W. E. L.","year":"1978","ISBN":"https:\/\/id.crossref.org\/isbn\/0919628206"},{"key":"21","doi-asserted-by":"crossref","first-page":"199","DOI":"10.24033\/bsmf.545","article-title":"Sur la distribution des z\u00e9ros de la fonction \ud835\udf01(\ud835\udc60) et ses cons\u00e9quences arithm\u00e9tiques","volume":"24","author":"Hadamard, J.","year":"1896","journal-title":"Bull. Soc. Math. France","ISSN":"https:\/\/id.crossref.org\/issn\/0037-9484","issn-type":"print"},{"key":"22","doi-asserted-by":"publisher","first-page":"33","DOI":"10.4153\/CMB-1972-007-7","article-title":"On the product of the primes","volume":"15","author":"Hanson, Denis","year":"1972","journal-title":"Canad. Math. Bull.","ISSN":"https:\/\/id.crossref.org\/issn\/0008-4395","issn-type":"print"},{"issue":"4","key":"23","doi-asserted-by":"publisher","first-page":"303","DOI":"10.4064\/aa117-4-1","article-title":"Une r\u00e9gion explicite sans z\u00e9ros pour la fonction \ud835\udf01 de Riemann","volume":"117","author":"Kadiri, Habiba","year":"2005","journal-title":"Acta Arith.","ISSN":"https:\/\/id.crossref.org\/issn\/0065-1036","issn-type":"print"},{"issue":"1","key":"24","doi-asserted-by":"publisher","first-page":"22","DOI":"10.1016\/j.jmaa.2018.04.071","article-title":"Explicit zero density for the Riemann zeta function","volume":"465","author":"Kadiri, Habiba","year":"2018","journal-title":"J. Math. Anal. Appl.","ISSN":"https:\/\/id.crossref.org\/issn\/0022-247X","issn-type":"print"},{"issue":"1","key":"25","doi-asserted-by":"publisher","first-page":"159","DOI":"10.1007\/BF02403071","article-title":"Sur la distribution des nombres premiers","volume":"24","author":"von Koch, Helge","year":"1901","journal-title":"Acta Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0001-5962","issn-type":"print"},{"key":"26","unstructured":"D. N. Lehmer, List of prime numbers from 1 to 10006721, Carnegie Institution of Washington, Washington, D.C., Publication no. 165, 1914."},{"issue":"2","key":"27","doi-asserted-by":"publisher","first-page":"372","DOI":"10.1137\/0519027","article-title":"Bounds for the tails of sharp-cutoff filter kernels","volume":"19","author":"Logan, B. F.","year":"1988","journal-title":"SIAM J. Math. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1410","issn-type":"print"},{"issue":"174","key":"28","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"},{"key":"29","first-page":"1","article-title":"The zeta function and prime numbers","author":"Montgomery, H. L.","year":"1980"},{"key":"30","doi-asserted-by":"publisher","first-page":"329","DOI":"10.1016\/j.jnt.2015.05.010","article-title":"Nonnegative trigonometric polynomials and a zero-free region for the Riemann zeta-function","volume":"157","author":"Mossinghoff, Michael J.","year":"2015","journal-title":"J. Number Theory","ISSN":"https:\/\/id.crossref.org\/issn\/0022-314X","issn-type":"print"},{"key":"31","doi-asserted-by":"publisher","first-page":"209","DOI":"10.4064\/aa-37-1-209-220","article-title":"On the remainder term of the prime number formula. II. On a theorem of Ingham","volume":"37","author":"Pintz, J.","year":"1980","journal-title":"Acta Arith.","ISSN":"https:\/\/id.crossref.org\/issn\/0065-1036","issn-type":"print"},{"issue":"293","key":"32","doi-asserted-by":"publisher","first-page":"1521","DOI":"10.1090\/S0025-5718-2014-02884-6","article-title":"Computing \ud835\udf0b(\ud835\udc65) analytically","volume":"84","author":"Platt, David J.","year":"2015","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"307","key":"33","doi-asserted-by":"publisher","first-page":"2449","DOI":"10.1090\/mcom\/3198","article-title":"Isolating some non-trivial zeros of zeta","volume":"86","author":"Platt, David J.","year":"2017","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"299","key":"34","doi-asserted-by":"publisher","first-page":"1539","DOI":"10.1090\/mcom\/3021","article-title":"On the first sign change of \ud835\udf03(\ud835\udc65)-\ud835\udc65","volume":"85","author":"Platt, D. J.","year":"2016","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"328","key":"35","doi-asserted-by":"publisher","first-page":"871","DOI":"10.1090\/mcom\/3583","article-title":"The error term in the prime number theorem","volume":"90","author":"Platt, David J.","year":"2021","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"key":"36","doi-asserted-by":"crossref","unstructured":"D. Platt and T. Trudgian, The Riemann hypothesis is true up to 3\u22c510\u00b9\u00b2, to appear in Bull. Lond. Math. Soc., \\url{https:\/\/doi.org\/10.1112\/blms.12460}.","DOI":"10.1112\/blms.12460"},{"issue":"297","key":"37","doi-asserted-by":"publisher","first-page":"325","DOI":"10.1090\/mcom\/2991","article-title":"An explicit density estimate for Dirichlet \ud835\udc3f-series","volume":"85","author":"Ramar\u00e9, O.","year":"2016","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"key":"38","doi-asserted-by":"publisher","first-page":"211","DOI":"10.2307\/2371291","article-title":"Explicit bounds for some functions of prime numbers","volume":"63","author":"Rosser, Barkley","year":"1941","journal-title":"Amer. J. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0002-9327","issn-type":"print"},{"key":"39","doi-asserted-by":"publisher","first-page":"595","DOI":"10.1512\/iumj.1955.4.54021","article-title":"Explicit remainder terms for some asymptotic series","volume":"4","author":"Rosser, J. Barkley","year":"1955","journal-title":"J. Rational Mech. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/1943-5282","issn-type":"print"},{"key":"40","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":"41","doi-asserted-by":"publisher","first-page":"243","DOI":"10.2307\/2005479","article-title":"Sharper bounds for the Chebyshev functions \ud835\udf03(\ud835\udc65) and \ud835\udf13(\ud835\udc65)","volume":"29","author":"Rosser, J. Barkley","year":"1975","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"295","key":"42","doi-asserted-by":"publisher","first-page":"2433","DOI":"10.1090\/S0025-5718-2015-02930-5","article-title":"A still sharper region where \ud835\udf0b(\ud835\udc65)-\ud835\udc59\ud835\udc56(\ud835\udc65) is positive","volume":"84","author":"Saouter, Yannick","year":"2015","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"134","key":"43","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"},{"key":"44","first-page":"419","article-title":"The zeros of the Riemann zeta-function","volume":"8","author":"Ste\u010dkin, S. B.","year":"1970","journal-title":"Mat. Zametki","ISSN":"https:\/\/id.crossref.org\/issn\/0025-567X","issn-type":"print"},{"issue":"2","key":"45","doi-asserted-by":"publisher","first-page":"225","DOI":"10.1007\/s11139-014-9656-6","article-title":"Updating the error term in the prime number theorem","volume":"39","author":"Trudgian, Tim","year":"2016","journal-title":"Ramanujan J.","ISSN":"https:\/\/id.crossref.org\/issn\/1382-4090","issn-type":"print"},{"key":"46","unstructured":"C. J. de la Vall\u00e9e Poussin, Recherches analytiques sur la th\u00e9orie des nombres premiers, Ann. Soc. Sci. Bruxelles, deuxi\u00e9me partie 20 (1896), 183\u2013256"},{"key":"47","unstructured":"S. Wedeniwski, ZETAGRID, computational verification of the Riemann hypothesis, Conference in Number Theory in Honour of Professor H.C. Williams, Alberta, Canada, May 2003, \\url{http:\/\/www.zetagrid.net\/zeta\/math\/zeta.result.100billion.zeros.html}."}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.ams.org\/mcom\/2021-90-331\/S0025-5718-2021-03643-1\/mcom3643_AM.pdf","content-type":"application\/pdf","content-version":"am","intended-application":"syndication"},{"URL":"https:\/\/www.ams.org\/mcom\/earlyview\/#mcom3643\/.pdf","content-type":"unspecified","content-version":"am","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/mcom\/2021-90-331\/S0025-5718-2021-03643-1\/S0025-5718-2021-03643-1.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,22]],"date-time":"2026-04-22T04:12:58Z","timestamp":1776831178000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/2021-90-331\/S0025-5718-2021-03643-1\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,6,15]]},"references-count":47,"journal-issue":{"issue":"331","published-print":{"date-parts":[[2021,9]]}},"alternative-id":["S0025-5718-2021-03643-1"],"URL":"https:\/\/doi.org\/10.1090\/mcom\/3643","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":[[2021,6,15]]}}}