{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,22]],"date-time":"2026-04-22T05:04:06Z","timestamp":1776834246841,"version":"3.51.2"},"reference-count":16,"publisher":"American Mathematical Society (AMS)","issue":"334","license":[{"start":{"date-parts":[[2022,10,15]],"date-time":"2022-10-15T00:00:00Z","timestamp":1665792000000},"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                    In a seminal paper, Robin proved that the Riemann hypothesis holds if and only if the inequality\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"sigma left-parenthesis n right-parenthesis equals sigma-summation Underscript d vertical-bar n Endscripts d greater-than e Superscript gamma Baseline n log log n\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>\n                              \u03c3\n                              \n                            <\/mml:mi>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>n<\/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>d<\/mml:mi>\n                                <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                  <mml:mo stretchy=\"false\">|<\/mml:mo>\n                                <\/mml:mrow>\n                                <mml:mi>n<\/mml:mi>\n                              <\/mml:mrow>\n                            <\/mml:munder>\n                            <mml:mi>d<\/mml:mi>\n                            <mml:mo>&gt;<\/mml:mo>\n                            <mml:msup>\n                              <mml:mi>e<\/mml:mi>\n                              <mml:mi>\n                                \u03b3\n                                \n                              <\/mml:mi>\n                            <\/mml:msup>\n                            <mml:mi>n<\/mml:mi>\n                            <mml:mi>log<\/mml:mi>\n                            <mml:mo>\n                              \u2061\n                              \n                            <\/mml:mo>\n                            <mml:mi>log<\/mml:mi>\n                            <mml:mo>\n                              \u2061\n                              \n                            <\/mml:mo>\n                            <mml:mi>n<\/mml:mi>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\sigma (n)=\\sum _{d|n} d &gt; e^\\gamma n \\log \\log n<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    holds for all integers greater than 5040, where\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"gamma\">\n                        <mml:semantics>\n                          <mml:mi>\n                            \u03b3\n                            \n                          <\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">\\gamma<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    is the Euler-Mascheroni constant. In this article, we prove new results on putative violations of Robin\u2019s criterion.\n                  <\/p>","DOI":"10.1090\/mcom\/3687","type":"journal-article","created":{"date-parts":[[2021,9,9]],"date-time":"2021-09-09T09:26:11Z","timestamp":1631179571000},"page":"909-920","source":"Crossref","is-referenced-by-count":0,"title":["New results for witnesses of Robin\u2019s criterion"],"prefix":"10.1090","volume":"91","author":[{"given":"Yannick","family":"Saouter","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"14","published-online":{"date-parts":[[2021,10,15]]},"reference":[{"issue":"2","key":"1","first-page":"187","article-title":"Grandes valeurs de la fonction somme des diviseurs et hypoth\u00e8se de Riemann","volume":"63","author":"Robin, G.","year":"1984","journal-title":"J. Math. Pures Appl. (9)","ISSN":"https:\/\/id.crossref.org\/issn\/0021-7824","issn-type":"print"},{"issue":"2","key":"2","doi-asserted-by":"publisher","first-page":"357","DOI":"10.5802\/jtnb.591","article-title":"On Robin\u2019s criterion for the Riemann hypothesis","volume":"19","author":"Choie, YoungJu","year":"2007","journal-title":"J. Th\\'{e}or. Nombres Bordeaux","ISSN":"https:\/\/id.crossref.org\/issn\/1246-7405","issn-type":"print"},{"issue":"4","key":"3","doi-asserted-by":"publisher","first-page":"303","DOI":"10.1007\/s00605-008-0022-x","article-title":"The Nicolas and Robin inequalities with sums of two squares","volume":"157","author":"Banks, William D.","year":"2009","journal-title":"Monatsh. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0026-9255","issn-type":"print"},{"issue":"2","key":"4","doi-asserted-by":"publisher","first-page":"301","DOI":"10.1515\/integ.2011.103","article-title":"The Robin inequality for 7-free integers","volume":"12","author":"Sol\u00e9, Patrick","year":"2012","journal-title":"Integers","ISSN":"https:\/\/id.crossref.org\/issn\/1867-0652","issn-type":"print"},{"key":"5","first-page":"Paper No. A12, 5","article-title":"Robin\u2019s inequality for 11-free integers","volume":"15","author":"Broughan, Kevin A.","year":"2015","journal-title":"Integers"},{"key":"6","first-page":"Paper No. A28, 7","article-title":"Robin\u2019s inequality for 20-free integers","volume":"21","author":"Morrill, Thomas","year":"2021","journal-title":"Integers"},{"issue":"2","key":"7","doi-asserted-by":"crossref","first-page":"251","DOI":"10.1080\/10586458.2006.10128957","article-title":"Abundant numbers and the Riemann hypothesis","volume":"15","author":"Briggs, Keith","year":"2006","journal-title":"Experiment. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/1058-6458","issn-type":"print"},{"issue":"1","key":"8","doi-asserted-by":"publisher","first-page":"227","DOI":"10.1007\/s11139-016-9839-4","article-title":"Explicit estimates of some functions over primes","volume":"45","author":"Dusart, Pierre","year":"2018","journal-title":"Ramanujan J.","ISSN":"https:\/\/id.crossref.org\/issn\/1382-4090","issn-type":"print"},{"issue":"170","key":"9","doi-asserted-by":"publisher","first-page":"537","DOI":"10.2307\/2007973","article-title":"Computing \ud835\udf0b(\ud835\udc65): the Meissel-Lehmer method","volume":"44","author":"Lagarias, J. C.","year":"1985","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"268","key":"10","doi-asserted-by":"publisher","first-page":"2283","DOI":"10.1090\/S0025-5718-09-02249-2","article-title":"Computing prime harmonic sums","volume":"78","author":"Bach, Eric","year":"2009","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"4","key":"11","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"},{"issue":"246","key":"12","doi-asserted-by":"publisher","first-page":"1023","DOI":"10.1090\/S0025-5718-03-01501-1","article-title":"Prime sieves using binary quadratic forms","volume":"73","author":"Atkin, A. O. L.","year":"2004","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"2","key":"13","doi-asserted-by":"publisher","first-page":"119","DOI":"10.1023\/A:1009764017495","article-title":"Highly composite numbers","volume":"1","author":"Ramanujan, Srinivasa","year":"1997","journal-title":"Ramanujan J.","ISSN":"https:\/\/id.crossref.org\/issn\/1382-4090","issn-type":"print"},{"issue":"301","key":"14","doi-asserted-by":"publisher","first-page":"2483","DOI":"10.1090\/mcom\/3060","article-title":"Estimating \ud835\udf0b(\ud835\udc65) and related functions under partial RH assumptions","volume":"85","author":"B\u00fcthe, Jan","year":"2016","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"3","key":"15","doi-asserted-by":"publisher","first-page":"792","DOI":"10.1112\/blms.12460","article-title":"The Riemann hypothesis is true up to 3\u22c510\u00b9\u00b2","volume":"53","author":"Platt, Dave","year":"2021","journal-title":"Bull. Lond. Math. Soc.","ISSN":"https:\/\/id.crossref.org\/issn\/0024-6093","issn-type":"print"},{"issue":"1","key":"16","doi-asserted-by":"publisher","first-page":"22","DOI":"10.1016\/j.jmaa.2018.04.071","article-title":"Explicit zero density for the Riemann zeta function","volume":"465","author":"Kadiri, Habiba","year":"2018","journal-title":"J. Math. Anal. Appl.","ISSN":"https:\/\/id.crossref.org\/issn\/0022-247X","issn-type":"print"}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.ams.org\/mcom\/2022-91-334\/S0025-5718-2021-03687-X\/S0025-5718-2021-03687-X.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,22]],"date-time":"2026-04-22T04:25:35Z","timestamp":1776831935000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/2022-91-334\/S0025-5718-2021-03687-X\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,10,15]]},"references-count":16,"journal-issue":{"issue":"334","published-print":{"date-parts":[[2022,3]]}},"alternative-id":["S0025-5718-2021-03687-X"],"URL":"https:\/\/doi.org\/10.1090\/mcom\/3687","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":[[2021,10,15]]}}}