{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,22]],"date-time":"2026-04-22T14:12:43Z","timestamp":1776867163178,"version":"3.51.2"},"reference-count":18,"publisher":"American Mathematical Society (AMS)","issue":"331","license":[{"start":{"date-parts":[[2022,4,2]],"date-time":"2022-04-02T00:00:00Z","timestamp":1648857600000},"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                    Assuming the generalized Riemann hypothesis, we give asymptotic bounds on the size of intervals that contain primes from a given arithmetic progression using the approach developed by Carneiro, Milinovich and Soundararajan [Comment. Math. Helv. 94, no. 3 (2019)]. For this we extend the Guinand-Weil explicit formula over all Dirichlet characters modulo\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"q greater-than-or-equal-to 3\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>q<\/mml:mi>\n                            <mml:mo>\n                              \u2265\n                              \n                            <\/mml:mo>\n                            <mml:mn>3<\/mml:mn>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">q \\geq 3<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    , and we reduce the associated extremal problems to convex optimization problems that can be solved numerically via semidefinite programming.\n                  <\/p>","DOI":"10.1090\/mcom\/3638","type":"journal-article","created":{"date-parts":[[2021,2,3]],"date-time":"2021-02-03T11:25:24Z","timestamp":1612351524000},"page":"2235-2246","source":"Crossref","is-referenced-by-count":2,"title":["Primes in arithmetic progressions and semidefinite programming"],"prefix":"10.1090","volume":"90","author":[{"given":"Andr\u00e9s","family":"Chirre","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Valdir","family":"Pereira J\u00fanior","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"David","family":"de Laat","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"14","published-online":{"date-parts":[[2021,4,2]]},"reference":[{"issue":"1","key":"1","doi-asserted-by":"publisher","first-page":"65","DOI":"10.1137\/141000671","article-title":"Julia: a fresh approach to numerical computing","volume":"59","author":"Bezanson, Jeff","year":"2017","journal-title":"SIAM Rev.","ISSN":"https:\/\/id.crossref.org\/issn\/1095-7200","issn-type":"print"},{"issue":"2","key":"2","doi-asserted-by":"publisher","first-page":"601","DOI":"10.5565\/PUBLMAT6321906","article-title":"Bandlimited approximations and estimates for the Riemann zeta-function","volume":"63","author":"Carneiro, Emanuel","year":"2019","journal-title":"Publ. 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Fys. 15 (1920), 1\u201333."},{"key":"8","series-title":"Graduate Texts in Mathematics","isbn-type":"print","doi-asserted-by":"crossref","DOI":"10.1007\/978-1-4757-5927-3","volume-title":"Multiplicative number theory","volume":"74","author":"Davenport, Harold","year":"1980","ISBN":"https:\/\/id.crossref.org\/isbn\/0387905332","edition":"2"},{"issue":"3","key":"9","doi-asserted-by":"publisher","first-page":"771","DOI":"10.1142\/S1793042115500426","article-title":"On the Riemann hypothesis and the difference between primes","volume":"11","author":"Dudek, Adrian W.","year":"2015","journal-title":"Int. J. Number Theory","ISSN":"https:\/\/id.crossref.org\/issn\/1793-0421","issn-type":"print"},{"issue":"5","key":"10","doi-asserted-by":"publisher","first-page":"1391","DOI":"10.1142\/S1793042116500858","article-title":"Primes in explicit short intervals on RH","volume":"12","author":"Dudek, Adrian W.","year":"2016","journal-title":"Int. J. 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