{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,9,25]],"date-time":"2026-09-25T03:35:19Z","timestamp":1790307319942,"version":"4.1.0"},"reference-count":17,"publisher":"American Mathematical Society (AMS)","issue":"275","license":[{"start":{"date-parts":[[2012,1,25]],"date-time":"2012-01-25T00:00:00Z","timestamp":1327449600000},"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                    Let\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper K\">\n                        <mml:semantics>\n                          <mml:mi>K<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">K<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    be an algebraic number field. Assume that\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"zeta Subscript upper K Baseline left-parenthesis s right-parenthesis slash zeta left-parenthesis s right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:msub>\n                              <mml:mi>\n                                \u03b6\n                                \n                              <\/mml:mi>\n                              <mml:mi>K<\/mml:mi>\n                            <\/mml:msub>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>s<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mo>\/<\/mml:mo>\n                            <\/mml:mrow>\n                            <mml:mi>\n                              \u03b6\n                              \n                            <\/mml:mi>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>s<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\zeta _K(s)\/\\zeta (s)<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    is entire. We give an explicit upper bound for the residue at\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"s equals 1\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>s<\/mml:mi>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:mn>1<\/mml:mn>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">s=1<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    of the Dedekind zeta function\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"zeta Subscript upper K Baseline left-parenthesis s right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:msub>\n                              <mml:mi>\n                                \u03b6\n                                \n                              <\/mml:mi>\n                              <mml:mi>K<\/mml:mi>\n                            <\/mml:msub>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>s<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\zeta _K(s)<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    of\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper K\">\n                        <mml:semantics>\n                          <mml:mi>K<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">K<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    . We deduce explicit upper bounds on class numbers of cubic and quartic number fields.\n                  <\/p>","DOI":"10.1090\/s0025-5718-2011-02457-9","type":"journal-article","created":{"date-parts":[[2011,1,25]],"date-time":"2011-01-25T08:48:19Z","timestamp":1295945299000},"page":"1813-1822","source":"Crossref","is-referenced-by-count":6,"title":["Upper bounds for residues of Dedekind zeta functions and class numbers of cubic and quartic number fields"],"prefix":"10.1090","volume":"80","author":[{"given":"St\u00e9phane","family":"Louboutin","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"14","published-online":{"date-parts":[[2011,1,25]]},"reference":[{"key":"1","isbn-type":"print","doi-asserted-by":"publisher","first-page":"63","DOI":"10.1007\/BFb0099441","article-title":"Lower bounds for regulators","author":"Cusick, T. 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R. Acad. Sci. Paris S\\'{e}r. I Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0764-4442","issn-type":"print"},{"issue":"5","key":"6","first-page":"443","article-title":"Majorations explicites de |\ud835\udc3f(1,\ud835\udf12)|. II","volume":"323","author":"Louboutin, Stephane","year":"1996","journal-title":"C. R. Acad. Sci. Paris S\\'{e}r. I Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0764-4442","issn-type":"print"},{"issue":"2","key":"7","doi-asserted-by":"publisher","first-page":"263","DOI":"10.1006\/jnth.2000.2545","article-title":"Explicit bounds for residues of Dedekind zeta functions, values of \ud835\udc3f-functions at \ud835\udc60=1, and relative class numbers","volume":"85","author":"Louboutin, St\u00e9phane","year":"2000","journal-title":"J. Number Theory","ISSN":"https:\/\/id.crossref.org\/issn\/0022-314X","issn-type":"print"},{"issue":"6","key":"8","doi-asserted-by":"publisher","first-page":"1194","DOI":"10.4153\/CJM-2001-045-5","article-title":"Explicit upper bounds for residues of Dedekind zeta functions and values of \ud835\udc3f-functions at \ud835\udc60=1, and explicit lower bounds for relative class numbers of CM-fields","volume":"53","author":"Louboutin, St\u00e9phane","year":"2001","journal-title":"Canad. J. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0008-414X","issn-type":"print"},{"key":"9","unstructured":"[Lou05] S. Louboutin. Explicit upper bounds for the residues at \ud835\udc60=1 of the Dedekind zeta functions of some totally real number fields. S\u00e9minaires & Congr\u00e8s 11 (2005), 171\u2013178; (AGCT 2003) SMF."},{"key":"10","series-title":"Progress in Mathematics","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-0348-0274-1","volume-title":"Non-vanishing of $L$-functions and applications","volume":"157","author":"Murty, M. Ram","year":"1997","ISBN":"https:\/\/id.crossref.org\/isbn\/3764358017"},{"issue":"4","key":"11","doi-asserted-by":"publisher","first-page":"701","DOI":"10.2969\/jmsj\/03340701","article-title":"Majoration du nombre de classes d\u2019un corps cubique cyclique de conducteur premier","volume":"33","author":"Moser, C.","year":"1981","journal-title":"J. Math. Soc. Japan","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5645","issn-type":"print"},{"issue":"1","key":"12","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1006\/jnth.1996.0031","article-title":"Certain quartic fields with small regulators","volume":"57","author":"Nakamula, Ken","year":"1996","journal-title":"J. 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Nombres Bordeaux (2)","ISSN":"https:\/\/id.crossref.org\/issn\/0989-5558","issn-type":"print"},{"issue":"3","key":"14","doi-asserted-by":"publisher","first-page":"245","DOI":"10.4064\/aa100-3-2","article-title":"Approximate formulae for \ud835\udc3f(1,\ud835\udf12)","volume":"100","author":"Ramar\u00e9, Olivier","year":"2001","journal-title":"Acta Arith.","ISSN":"https:\/\/id.crossref.org\/issn\/0065-1036","issn-type":"print"},{"issue":"3","key":"15","doi-asserted-by":"publisher","first-page":"437","DOI":"10.1016\/0022-314X(84)90082-9","article-title":"An inequality relating the regulator and the discriminant of a number field","volume":"19","author":"Silverman, Joseph H.","year":"1984","journal-title":"J. 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