{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,22]],"date-time":"2026-04-22T03:07:45Z","timestamp":1776827265359,"version":"3.51.2"},"reference-count":10,"publisher":"American Mathematical Society (AMS)","issue":"239","license":[{"start":{"date-parts":[[2003,1,11]],"date-time":"2003-01-11T00:00:00Z","timestamp":1042243200000},"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>In this paper, we are interested in the evaluation of the zeta function of the simplest cubic field. We first introduce Siegel\u2019s formula for values of the zeta function of a totally real number field at negative odd integers. Next, we will develop a method of computing the sum of a divisor function for ideals, and will give a full description for a Siegel lattice of the simplest cubic field. Using these results, we will derive explicit expressions, which involve only rational integers, for values of a zeta function of the simplest cubic field. Finally, as an illustration of our method, we will give a table for zeta values for the first one hundred simplest cubic fields.<\/p>","DOI":"10.1090\/s0025-5718-02-01395-9","type":"journal-article","created":{"date-parts":[[2002,9,20]],"date-time":"2002-09-20T15:46:54Z","timestamp":1032536814000},"page":"1243-1262","source":"Crossref","is-referenced-by-count":5,"title":["Evaluation of zeta function of the simplest cubic field at negative odd integers"],"prefix":"10.1090","volume":"71","author":[{"given":"Hyun","family":"Kim","sequence":"first","affiliation":[]},{"given":"Jung","family":"Kim","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2002,1,11]]},"reference":[{"issue":"1","key":"1","first-page":"13","article-title":"Special values of zeta functions of the simplest cubic fields and their applications","volume":"74","author":"Byeon, Dongho","year":"1998","journal-title":"Proc. Japan Acad. Ser. A Math. Sci.","ISSN":"https:\/\/id.crossref.org\/issn\/0386-2194","issn-type":"print"},{"issue":"3","key":"2","doi-asserted-by":"publisher","first-page":"266","DOI":"10.1016\/0022-314X(90)90090-E","article-title":"On the computation of the values of zeta functions of totally real cubic fields","volume":"36","author":"Halbritter, U.","year":"1990","journal-title":"J. Number Theory","ISSN":"https:\/\/id.crossref.org\/issn\/0022-314X","issn-type":"print"},{"key":"3","doi-asserted-by":"crossref","unstructured":"H. K. Kim and H. J. Hwang, Values of zeta functions and class number 1 criterion for the simplest cubic fields, Nagoya Math. J. 160 (2000), 161-180.","DOI":"10.1017\/S0027763000007741"},{"key":"4","unstructured":"J. S. Kim, Determination of class numbers of the simplest cubic fields, to appear in Comm. of the Korean Math. Soc."},{"key":"5","unstructured":"A. J. Lazarus, The class number and cyclotomy of simplest quartic fields, PhD thesis, University of California, Berkeley, 1989."},{"key":"6","doi-asserted-by":"publisher","first-page":"1137","DOI":"10.2307\/2005372","article-title":"The simplest cubic fields","volume":"28","author":"Shanks, Daniel","year":"1974","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"key":"7","first-page":"87","article-title":"Berechnung von Zetafunktionen an ganzzahligen Stellen","volume":"1969","author":"Siegel, Carl Ludwig","year":"1969","journal-title":"Nachr. Akad. Wiss. G\\\"{o}ttingen Math.-Phys. Kl. II","ISSN":"https:\/\/id.crossref.org\/issn\/0065-5295","issn-type":"print"},{"issue":"177","key":"8","doi-asserted-by":"publisher","first-page":"371","DOI":"10.2307\/2007897","article-title":"Class numbers of the simplest cubic fields","volume":"48","author":"Washington, Lawrence C.","year":"1987","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"key":"9","series-title":"Graduate Texts in Mathematics","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4684-0133-2","volume-title":"Introduction to cyclotomic fields","volume":"83","author":"Washington, Lawrence C.","year":"1982","ISBN":"https:\/\/id.crossref.org\/isbn\/0387906223"},{"issue":"1-2","key":"10","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\/2002-71-239\/S0025-5718-02-01395-9\/S0025-5718-02-01395-9.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/mcom\/2002-71-239\/S0025-5718-02-01395-9\/S0025-5718-02-01395-9.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,20]],"date-time":"2026-04-20T23:00:45Z","timestamp":1776726045000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/2002-71-239\/S0025-5718-02-01395-9\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2002,1,11]]},"references-count":10,"journal-issue":{"issue":"239","published-print":{"date-parts":[[2002,7]]}},"alternative-id":["S0025-5718-02-01395-9"],"URL":"https:\/\/doi.org\/10.1090\/s0025-5718-02-01395-9","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":[[2002,1,11]]}}}