{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,22]],"date-time":"2026-04-22T08:34:23Z","timestamp":1776846863443,"version":"3.51.2"},"reference-count":13,"publisher":"American Mathematical Society (AMS)","issue":"294","license":[{"start":{"date-parts":[[2015,12,29]],"date-time":"2015-12-29T00:00:00Z","timestamp":1451347200000},"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                    This article considers linear relations between the nontrivial zeroes of the Riemann zeta-function. The main application is an alternative disproof of Mertens\u2019 conjecture by showing that\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"limit sup upper M left-parenthesis x right-parenthesis x Superscript negative 1 slash 2 Baseline greater-than-or-equal-to 1.6383\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:munder>\n                              <mml:mo movablelimits=\"true\" form=\"prefix\">lim\u2006sup<\/mml:mo>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mi>x<\/mml:mi>\n                                <mml:mo stretchy=\"false\">\n                                  \u2192\n                                  \n                                <\/mml:mo>\n                                <mml:mi mathvariant=\"normal\">\n                                  \u221e\n                                  \n                                <\/mml:mi>\n                              <\/mml:mrow>\n                            <\/mml:munder>\n                            <mml:mi>M<\/mml:mi>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>x<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                            <mml:msup>\n                              <mml:mi>x<\/mml:mi>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mo>\n                                  \u2212\n                                  \n                                <\/mml:mo>\n                                <mml:mn>1<\/mml:mn>\n                                <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                  <mml:mo>\/<\/mml:mo>\n                                <\/mml:mrow>\n                                <mml:mn>2<\/mml:mn>\n                              <\/mml:mrow>\n                            <\/mml:msup>\n                            <mml:mo>\n                              \u2265\n                              \n                            <\/mml:mo>\n                            <mml:mn>1.6383<\/mml:mn>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\limsup _{x\\rightarrow \\infty } M(x) x^{-1\/2} \\geq 1.6383<\/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=\"limit inf upper M left-parenthesis x right-parenthesis x Superscript negative 1 slash 2 Baseline less-than-or-equal-to negative 1.6383 period\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:munder>\n                              <mml:mo movablelimits=\"true\" form=\"prefix\">lim\u2006inf<\/mml:mo>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mi>x<\/mml:mi>\n                                <mml:mo stretchy=\"false\">\n                                  \u2192\n                                  \n                                <\/mml:mo>\n                                <mml:mi mathvariant=\"normal\">\n                                  \u221e\n                                  \n                                <\/mml:mi>\n                              <\/mml:mrow>\n                            <\/mml:munder>\n                            <mml:mi>M<\/mml:mi>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>x<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                            <mml:msup>\n                              <mml:mi>x<\/mml:mi>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mo>\n                                  \u2212\n                                  \n                                <\/mml:mo>\n                                <mml:mn>1<\/mml:mn>\n                                <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                  <mml:mo>\/<\/mml:mo>\n                                <\/mml:mrow>\n                                <mml:mn>2<\/mml:mn>\n                              <\/mml:mrow>\n                            <\/mml:msup>\n                            <mml:mo>\n                              \u2264\n                              \n                            <\/mml:mo>\n                            <mml:mo>\n                              \u2212\n                              \n                            <\/mml:mo>\n                            <mml:mn>1.6383.<\/mml:mn>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\liminf _{x\\rightarrow \\infty } M(x) x^{-1\/2} \\leq -1.6383.<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                  <\/p>","DOI":"10.1090\/s0025-5718-2014-02916-5","type":"journal-article","created":{"date-parts":[[2014,12,29]],"date-time":"2014-12-29T15:18:44Z","timestamp":1419866324000},"page":"2047-2058","source":"Crossref","is-referenced-by-count":11,"title":["Linear relations of zeroes of the zeta-function"],"prefix":"10.1090","volume":"84","author":[{"given":"D.","family":"Best","sequence":"first","affiliation":[]},{"given":"T.","family":"Trudgian","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2014,12,29]]},"reference":[{"key":"1","isbn-type":"print","first-page":"79","article-title":"Oscillation theorems","author":"Anderson, Robert J.","year":"1981","ISBN":"https:\/\/id.crossref.org\/isbn\/3540111735"},{"key":"2","first-page":"11","article-title":"Linear relations connecting the imaginary parts of the zeros of the zeta function","author":"Bateman, P. T.","year":"1971"},{"key":"3","series-title":"Pure and Applied Mathematics (Boca Raton)","isbn-type":"print","volume-title":"Lattice basis reduction","volume":"300","author":"Bremner, Murray R.","year":"2012","ISBN":"https:\/\/id.crossref.org\/isbn\/9781439807026"},{"key":"4","series-title":"Graduate Texts in Mathematics","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-662-02945-9","volume-title":"A course in computational algebraic number theory","volume":"138","author":"Cohen, Henri","year":"1993","ISBN":"https:\/\/id.crossref.org\/isbn\/3540556400"},{"key":"5","first-page":"113","article-title":"Two oscillation theorems","author":"Diamond, Harold G.","year":"1972"},{"key":"6","first-page":"141","article-title":"Oscillation theorems","author":"Grosswald, Emil","year":"1972"},{"key":"7","doi-asserted-by":"publisher","first-page":"141","DOI":"10.1112\/S0025579300001480","article-title":"A disproof of a conjecture of P\u00f3lya","volume":"5","author":"Haselgrove, C. B.","year":"1958","journal-title":"Mathematika","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5793","issn-type":"print"},{"issue":"1-2","key":"8","first-page":"47","article-title":"Numerical study of the derivative of the Riemann zeta function at zeros","volume":"60","author":"Hiary, Ghaith A.","year":"2011","journal-title":"Comment. Math. Univ. St. Pauli","ISSN":"https:\/\/id.crossref.org\/issn\/0010-258X","issn-type":"print"},{"key":"9","doi-asserted-by":"publisher","first-page":"313","DOI":"10.2307\/2371685","article-title":"On two conjectures in the theory of numbers","volume":"64","author":"Ingham, A. E.","year":"1942","journal-title":"Amer. J. 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Ann.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5831","issn-type":"print"},{"key":"13","doi-asserted-by":"publisher","first-page":"138","DOI":"10.1515\/crll.1985.357.138","article-title":"Disproof of the Mertens conjecture","volume":"357","author":"Odlyzko, A. M.","year":"1985","journal-title":"J. Reine Angew. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0075-4102","issn-type":"print"}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/www.ams.org\/mcom\/2015-84-294\/S0025-5718-2014-02916-5\/S0025-5718-2014-02916-5.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/mcom\/2015-84-294\/S0025-5718-2014-02916-5\/S0025-5718-2014-02916-5.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T18:24:38Z","timestamp":1776795878000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/2015-84-294\/S0025-5718-2014-02916-5\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2014,12,29]]},"references-count":13,"journal-issue":{"issue":"294","published-print":{"date-parts":[[2015,7]]}},"alternative-id":["S0025-5718-2014-02916-5"],"URL":"https:\/\/doi.org\/10.1090\/s0025-5718-2014-02916-5","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":[[2014,12,29]]}}}