{"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":1776846014120,"version":"3.51.2"},"reference-count":21,"publisher":"American Mathematical Society (AMS)","issue":"276","license":[{"start":{"date-parts":[[2012,3,1]],"date-time":"2012-03-01T00:00:00Z","timestamp":1330560000000},"content-version":"am","delay-in-days":366,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>\n                    This article improves the estimate of the size of the definite integral of\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper S left-parenthesis t right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>S<\/mml:mi>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>t<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">S(t)<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    , the argument of the Riemann zeta-function. The primary application of this improvement is Turing\u2019s Method for the Riemann zeta-function. Analogous improvements are given for the arguments of Dirichlet\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper L\">\n                        <mml:semantics>\n                          <mml:mi>L<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">L<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    -functions and of Dedekind zeta-functions.\n                  <\/p>","DOI":"10.1090\/s0025-5718-2011-02470-1","type":"journal-article","created":{"date-parts":[[2011,3,1]],"date-time":"2011-03-01T19:52:42Z","timestamp":1299009162000},"page":"2259-2279","source":"Crossref","is-referenced-by-count":12,"title":["Improvements to Turing\u2019s method"],"prefix":"10.1090","volume":"80","author":[{"given":"Timothy","family":"Trudgian","sequence":"first","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2011,3,1]]},"reference":[{"key":"1","unstructured":"R. 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Gram\u2019s Law fails a positive proportion of the time. arxiv.org\/abs\/0811.0883, 2008."},{"key":"20","doi-asserted-by":"publisher","first-page":"99","DOI":"10.1112\/plms\/s3-3.1.99","article-title":"Some calculations of the Riemann zeta-function","volume":"3","author":"Turing, A. M.","year":"1953","journal-title":"Proc. London Math. Soc. (3)","ISSN":"https:\/\/id.crossref.org\/issn\/0024-6115","issn-type":"print"},{"key":"21","series-title":"Cambridge Mathematical Library","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511608759","volume-title":"A course of modern analysis","author":"Whittaker, E. 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