{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,9,12]],"date-time":"2025-09-12T18:08:33Z","timestamp":1757700513214},"reference-count":17,"publisher":"Wiley","issue":"1","license":[{"start":{"date-parts":[[2016,2,1]],"date-time":"2016-02-01T00:00:00Z","timestamp":1454284800000},"content-version":"unspecified","delay-in-days":31,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["LMS J. Comput. Math."],"published-print":{"date-parts":[[2016]]},"abstract":"<jats:p>This article considers the positive integers <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S1461157015000340_inline2\" \/><jats:tex-math>$N$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> for which <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S1461157015000340_inline3\" \/><jats:tex-math>${\\it\\zeta}_{N}(s)=\\sum _{n=1}^{N}n^{-s}$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> has zeroes in the half-plane <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S1461157015000340_inline4\" \/><jats:tex-math>$\\Re (s)&gt;1$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>. Building on earlier results, we show that there are no zeroes for <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S1461157015000340_inline5\" \/><jats:tex-math>$1\\leqslant N\\leqslant 18$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> and for <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S1461157015000340_inline6\" \/><jats:tex-math>$N=20,21,28$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>. For all other <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S1461157015000340_inline7\" \/><jats:tex-math>$N$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> there are infinitely many such zeroes.<\/jats:p>","DOI":"10.1112\/s1461157015000340","type":"journal-article","created":{"date-parts":[[2016,2,29]],"date-time":"2016-02-29T06:32:03Z","timestamp":1456727523000},"page":"37-41","source":"Crossref","is-referenced-by-count":2,"title":["Zeroes of partial sums of the zeta-function"],"prefix":"10.1112","volume":"19","author":[{"given":"David J.","family":"Platt","sequence":"first","affiliation":[]},{"given":"Timothy S.","family":"Trudgian","sequence":"additional","affiliation":[]}],"member":"311","published-online":{"date-parts":[[2016,2,1]]},"reference":[{"key":"S1461157015000340_r14","doi-asserted-by":"publisher","DOI":"10.1215\/S0012-7094-68-03508-4"},{"key":"S1461157015000340_r1","unstructured":"1. 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IMRN"}],"container-title":["LMS Journal of Computation and Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S1461157015000340","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,4,19]],"date-time":"2019-04-19T20:09:42Z","timestamp":1555704582000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S1461157015000340\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2016]]},"references-count":17,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2016]]}},"alternative-id":["S1461157015000340"],"URL":"https:\/\/doi.org\/10.1112\/s1461157015000340","relation":{},"ISSN":["1461-1570"],"issn-type":[{"value":"1461-1570","type":"electronic"}],"subject":[],"published":{"date-parts":[[2016]]}}}