{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T11:05:12Z","timestamp":1776769512791,"version":"3.51.2"},"reference-count":11,"publisher":"American Mathematical Society (AMS)","issue":"238","license":[{"start":{"date-parts":[[2002,8,3]],"date-time":"2002-08-03T00:00:00Z","timestamp":1028332800000},"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 previous work, the author has extended the concept of regular and irregular primes to the setting of arbitrary totally real number fields\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"k 0\">\n                        <mml:semantics>\n                          <mml:msub>\n                            <mml:mi>k<\/mml:mi>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mn>0<\/mml:mn>\n                            <\/mml:mrow>\n                          <\/mml:msub>\n                          <mml:annotation encoding=\"application\/x-tex\">k_{0}<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    , using the values of the zeta function\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"zeta Subscript k 0\">\n                        <mml:semantics>\n                          <mml:msub>\n                            <mml:mi>\n                              \u03b6\n                              \n                            <\/mml:mi>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:msub>\n                                <mml:mi>k<\/mml:mi>\n                                <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                  <mml:mn>0<\/mml:mn>\n                                <\/mml:mrow>\n                              <\/mml:msub>\n                            <\/mml:mrow>\n                          <\/mml:msub>\n                          <mml:annotation encoding=\"application\/x-tex\">\\zeta _{k_{0}}<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    at negative integers as our \u201chigher Bernoulli numbers\u201d. In the case where\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"k 0\">\n                        <mml:semantics>\n                          <mml:msub>\n                            <mml:mi>k<\/mml:mi>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mn>0<\/mml:mn>\n                            <\/mml:mrow>\n                          <\/mml:msub>\n                          <mml:annotation encoding=\"application\/x-tex\">k_{0}<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    is a real quadratic field, Siegel presented two formulas for calculating these zeta-values: one using entirely elementary methods and one which is derived from the theory of modular forms. (The author would like to thank Henri Cohen for suggesting an analysis of the second formula.) We briefly discuss several algorithms based on these formulas and compare the running time involved in using them to determine the index of\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"k 0\">\n                        <mml:semantics>\n                          <mml:msub>\n                            <mml:mi>k<\/mml:mi>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mn>0<\/mml:mn>\n                            <\/mml:mrow>\n                          <\/mml:msub>\n                          <mml:annotation encoding=\"application\/x-tex\">k_{0}<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    -irregularity (more generally, \u201cquadratic irregularity\u201d) of a prime number.\n                  <\/p>","DOI":"10.1090\/s0025-5718-01-01341-2","type":"journal-article","created":{"date-parts":[[2002,7,26]],"date-time":"2002-07-26T18:14:28Z","timestamp":1027707268000},"page":"863-871","source":"Crossref","is-referenced-by-count":1,"title":["Comparison of algorithms to calculate quadratic irregularity of prime numbers"],"prefix":"10.1090","volume":"71","author":[{"given":"Joshua","family":"Holden","sequence":"first","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2001,8,3]]},"reference":[{"key":"1","series-title":"Undergraduate Texts in Mathematics","volume-title":"Introduction to analytic number theory","author":"Apostol, Tom M.","year":"1976"},{"key":"2","unstructured":"C. Batut, K. Belabas, D. Bernardi, H. Cohen, and M. Olivier, User\u2019s guide to PARI-GP, Laboratoire A2X, Universit\u00e9 Bordeaux I, version 2.0.9 ed., May 13, 1998, \\url{http:\/\/hasse.mathematik.tu-muenchen.de\/ntsw\/pari\/Welcome.html}, \\url{ftp:\/\/megrez.-math.u-bordeaux.fr}."},{"key":"3","doi-asserted-by":"crossref","unstructured":"Johannes Buchmann and Sachar Paulus, A one way function based on ideal arithmetic in number fields, Advances in cryptology\u2014CRYPTO \u201997 (Burton S. Kaliski, Jr, ed.), Lecture Notes in Computer Science, vol. 1294, Springer-Verlag, 1997, pp. 385\u2013394.","DOI":"10.1007\/BFb0052250"},{"key":"4","first-page":"Exp. No. 3, 21","article-title":"Sums involving the values at negative integers of \ud835\udc3f functions of quadratic characters","author":"Cohen, Henri","year":"1975"},{"issue":"1","key":"5","doi-asserted-by":"publisher","first-page":"63","DOI":"10.4064\/aa-30-1-63-93","article-title":"Variations sur un th\u00e8me de Seigel et Hecke","volume":"30","author":"Cohen, Henri","year":"1976","journal-title":"Acta Arith.","ISSN":"https:\/\/id.crossref.org\/issn\/0065-1036","issn-type":"print"},{"key":"6","isbn-type":"print","doi-asserted-by":"publisher","first-page":"454","DOI":"10.1007\/BFb0054884","article-title":"Irregularity of prime numbers over real quadratic fields","author":"Holden, Joshua","year":"1998","ISBN":"https:\/\/id.crossref.org\/isbn\/3540646574"},{"key":"7","unstructured":"\\bysame, On the Fontaine-Mazur conjecture for number fields and an analogue for function fields, Ph.D. thesis, Brown University, 1998."},{"issue":"1","key":"8","doi-asserted-by":"publisher","first-page":"16","DOI":"10.1006\/jnth.1999.2434","article-title":"On the Fontaine-Mazur conjecture for number fields and an analogue for function fields","volume":"81","author":"Holden, Joshua Brandon","year":"2000","journal-title":"J. Number Theory","ISSN":"https:\/\/id.crossref.org\/issn\/0022-314X","issn-type":"print"},{"key":"9","unstructured":"\\bysame, First-hit analysis of algorithms for computing quadratic irregularity, (In preparation)."},{"key":"10","first-page":"7","article-title":"Bernoullische Polynome und quadratische Zahlk\u00f6rper","volume":"1968","author":"Siegel, Carl Ludwig","year":"1968","journal-title":"Nachr. Akad. Wiss. G\\\"{o}ttingen Math.-Phys. Kl. II","ISSN":"https:\/\/id.crossref.org\/issn\/0065-5295","issn-type":"print"},{"issue":"1-2","key":"11","first-page":"55","article-title":"On the values at negative integers of the zeta-function of a real quadratic field","volume":"22","author":"Zagier, Don","year":"1976","journal-title":"Enseign. Math. (2)","ISSN":"https:\/\/id.crossref.org\/issn\/0013-8584","issn-type":"print"}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/www.ams.org\/mcom\/2002-71-238\/S0025-5718-01-01341-2\/S0025-5718-01-01341-2.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/mcom\/2002-71-238\/S0025-5718-01-01341-2\/S0025-5718-01-01341-2.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,20]],"date-time":"2026-04-20T22:56:58Z","timestamp":1776725818000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/2002-71-238\/S0025-5718-01-01341-2\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2001,8,3]]},"references-count":11,"journal-issue":{"issue":"238","published-print":{"date-parts":[[2002,4]]}},"alternative-id":["S0025-5718-01-01341-2"],"URL":"https:\/\/doi.org\/10.1090\/s0025-5718-01-01341-2","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":[[2001,8,3]]}}}