{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,22]],"date-time":"2026-04-22T08:42:24Z","timestamp":1776847344331,"version":"3.51.2"},"reference-count":21,"publisher":"American Mathematical Society (AMS)","issue":"299","license":[{"start":{"date-parts":[[2016,8,21]],"date-time":"2016-08-21T00:00:00Z","timestamp":1471737600000},"content-version":"am","delay-in-days":366,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"funder":[{"DOI":"10.13039\/501100000923","name":"Australian Research Council","doi-asserted-by":"publisher","award":["DE120100173"],"award-info":[{"award-number":["DE120100173"]}],"id":[{"id":"10.13039\/501100000923","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>\n                    Let\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 equals sigma-summation Underscript p less-than-or-equal-to x Endscripts log p\">\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>=<\/mml:mo>\n                            <mml:munder>\n                              <mml:mo>\n                                \u2211\n                                \n                              <\/mml:mo>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mi>p<\/mml:mi>\n                                <mml:mo>\n                                  \u2264\n                                  \n                                <\/mml:mo>\n                                <mml:mi>x<\/mml:mi>\n                              <\/mml:mrow>\n                            <\/mml:munder>\n                            <mml:mi>log<\/mml:mi>\n                            <mml:mo>\n                              \u2061\n                              \n                            <\/mml:mo>\n                            <mml:mi>p<\/mml:mi>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\theta (x) = \\sum _{p\\leq x} \\log p<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    . We show that\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 greater-than 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>&gt;<\/mml:mo>\n                            <mml:mi>x<\/mml:mi>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\theta (x)&gt;x<\/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=\"2 greater-than x greater-than 1.39 dot 10 Superscript 17\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mn>2<\/mml:mn>\n                            <mml:mo>&gt;<\/mml:mo>\n                            <mml:mi>x<\/mml:mi>\n                            <mml:mo>&gt;<\/mml:mo>\n                            <mml:mn>1.39<\/mml:mn>\n                            <mml:mo>\n                              \u22c5\n                              \n                            <\/mml:mo>\n                            <mml:msup>\n                              <mml:mn>10<\/mml:mn>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mn>17<\/mml:mn>\n                              <\/mml:mrow>\n                            <\/mml:msup>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">2&gt;x&gt; 1.39\\cdot 10^{17}<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    . We also show that there is an\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"x greater-than exp left-parenthesis 727.951332668 right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>x<\/mml:mi>\n                            <mml:mo>&gt;<\/mml:mo>\n                            <mml:mi>exp<\/mml:mi>\n                            <mml:mo>\n                              \u2061\n                              \n                            <\/mml:mo>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mn>727.951332668<\/mml:mn>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">x&gt;\\exp (727.951332668)<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    for which\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 greater-than x period\">\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>&gt;<\/mml:mo>\n                            <mml:mi>x<\/mml:mi>\n                            <mml:mo>.<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\theta (x) &gt;x.<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                  <\/p>","DOI":"10.1090\/mcom\/3021","type":"journal-article","created":{"date-parts":[[2015,8,21]],"date-time":"2015-08-21T08:17:25Z","timestamp":1440145045000},"page":"1539-1547","source":"Crossref","is-referenced-by-count":20,"title":["On the first sign change of \ud835\udf03(\ud835\udc65)-\ud835\udc65"],"prefix":"10.1090","volume":"85","author":[{"given":"D.","family":"Platt","sequence":"first","affiliation":[]},{"given":"T.","family":"Trudgian","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2015,8,21]]},"reference":[{"key":"1","unstructured":"ACRC, Bluecrystal phase 3 user guide, 2014."},{"issue":"231","key":"2","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":"3","unstructured":"J. Bober, Database of zeros of the zeta function, 2012. \\url{http:\/\/sage.math.washington.edu\/home\/bober\/www\/data\/platt_{z}eros\/zeros}."},{"issue":"3","key":"4","doi-asserted-by":"publisher","first-page":"289","DOI":"10.1080\/10586458.2014.990118","article-title":"On solving a curious inequality of Ramanujan","volume":"24","author":"Dudek, Adrian W.","year":"2015","journal-title":"Exp. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/1058-6458","issn-type":"print"},{"key":"5","unstructured":"P. Dusart, Estimates of some functions over primes without R.H., arXiv:1002.0442v1, 2010."},{"issue":"293","key":"6","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"},{"key":"7","series-title":"Cambridge Mathematical Library","isbn-type":"print","volume-title":"The distribution of prime numbers","author":"Ingham, A. E.","year":"1990","ISBN":"https:\/\/id.crossref.org\/isbn\/0521397898"},{"issue":"1","key":"8","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":"9","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":"10","unstructured":"J. E. Littlewood, Sur la distribution des nombres premiers, Comptes Rendus, 158 (1914), 1869\u20131872."},{"key":"11","doi-asserted-by":"publisher","first-page":"235","DOI":"10.7169\/facm\/1229442626","article-title":"Changes of sign of the error term in the prime number theorem","volume":"35","author":"Montgomery, Hugh L.","year":"2006","journal-title":"Funct. Approx. Comment. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0208-6573","issn-type":"print"},{"key":"12","unstructured":"T. Oliveira e Silva, Tables of values of pi(x) and pi2(x), 2012, \\url{http:\/\/www.ieeta.pt\/ tos\/primes.html}."},{"key":"13","unstructured":"D. J. Platt. Computing Degree 1 \ud835\udc3f-functions Rigorously. PhD thesis, Bristol University, 2011."},{"issue":"4","key":"14","doi-asserted-by":"publisher","first-page":"275","DOI":"10.1007\/s11155-005-6891-y","article-title":"Motivations for an arbitrary precision interval arithmetic and the MPFI library","volume":"11","author":"Revol, Nathalie","year":"2005","journal-title":"Reliab. Comput.","ISSN":"https:\/\/id.crossref.org\/issn\/1385-3139","issn-type":"print"},{"key":"15","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":"16","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":"295","key":"17","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":"18","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":"19","doi-asserted-by":"publisher","first-page":"48","DOI":"10.1112\/plms\/s3-5.1.48","article-title":"On the difference \ud835\udf0b(\ud835\udc65)-\ud835\udc59\ud835\udc56\ud835\udc65. II","volume":"5","author":"Skewes, S.","year":"1955","journal-title":"Proc. London Math. Soc. (3)","ISSN":"https:\/\/id.crossref.org\/issn\/0024-6115","issn-type":"print"},{"key":"20","unstructured":"T. S. Trudgian, Updating the error term in the prime number theorem, Ramanujan J, 2014. To appear. Preprint available at arXiv:1401.2689v1."},{"key":"21","unstructured":"K. Walisch, Primesieve, 2012. \\url{http:\/\/code.google.com\/p\/primesieve\/}."}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/www.ams.org\/mcom\/2016-85-299\/S0025-5718-2015-03021-X\/S0025-5718-2015-03021-X.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/mcom\/2016-85-299\/S0025-5718-2015-03021-X\/S0025-5718-2015-03021-X.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T18:45:56Z","timestamp":1776797156000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/2016-85-299\/S0025-5718-2015-03021-X\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2015,8,21]]},"references-count":21,"journal-issue":{"issue":"299","published-print":{"date-parts":[[2016,5]]}},"alternative-id":["S0025-5718-2015-03021-X"],"URL":"https:\/\/doi.org\/10.1090\/mcom\/3021","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":[[2015,8,21]]}}}