{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,22]],"date-time":"2026-04-22T07:59:59Z","timestamp":1776844799451,"version":"3.51.2"},"reference-count":14,"publisher":"American Mathematical Society (AMS)","issue":"299","license":[{"start":{"date-parts":[[2016,10,9]],"date-time":"2016-10-09T00:00:00Z","timestamp":1475971200000},"content-version":"am","delay-in-days":366,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>\n                    Assuming GRH, we prove an explicit upper bound for the number of zeros of a Dedekind zeta function having imaginary part in\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"left-bracket upper T minus a comma upper T plus a right-bracket\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mo stretchy=\"false\">[<\/mml:mo>\n                            <mml:mi>T<\/mml:mi>\n                            <mml:mo>\n                              \u2212\n                              \n                            <\/mml:mo>\n                            <mml:mi>a<\/mml:mi>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mi>T<\/mml:mi>\n                            <mml:mo>+<\/mml:mo>\n                            <mml:mi>a<\/mml:mi>\n                            <mml:mo stretchy=\"false\">]<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">[T-a,T+a]<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    . We also prove a bound for the multiplicity of the zeros.\n                  <\/p>","DOI":"10.1090\/mcom\/3024","type":"journal-article","created":{"date-parts":[[2015,10,9]],"date-time":"2015-10-09T08:39:01Z","timestamp":1444379941000},"page":"1503-1522","source":"Crossref","is-referenced-by-count":1,"title":["Zeros of Dedekind zeta functions under GRH"],"prefix":"10.1090","volume":"85","author":[{"given":"Lo\u00efc","family":"Greni\u00e9","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Giuseppe","family":"Molteni","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"14","published-online":{"date-parts":[[2015,10,9]]},"reference":[{"key":"1","series-title":"Encyclopedia of Mathematics and its Applications","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9781107325937","volume-title":"Special functions","volume":"71","author":"Andrews, George E.","year":"1999","ISBN":"https:\/\/id.crossref.org\/isbn\/0521623219"},{"key":"2","unstructured":"J. W. Bober, Database of zeros of the zeta function, \\url{http:\/\/www.lmfdb.org\/zeros\/zeta\/}, 2014."},{"issue":"3","key":"3","doi-asserted-by":"publisher","first-page":"939","DOI":"10.1007\/s00208-012-0876-z","article-title":"Bounding \ud835\udc46(\ud835\udc61) and \ud835\udc46\u2081(\ud835\udc61) on the Riemann hypothesis","volume":"356","author":"Carneiro, Emanuel","year":"2013","journal-title":"Math. Ann.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5831","issn-type":"print"},{"key":"4","unstructured":"X. Gourdon, The 10\u00b9\u00b3 first zeros of the Riemann zeta function, and zeros computation at very large height, 2004, \\url{http:\/\/numbers.computation.free.fr\/Constants\/Miscellaneous\/} zetazeros1e13-1e24.pdf"},{"key":"5","doi-asserted-by":"crossref","unstructured":"L. Greni\u00e9 and G. Molteni, Explicit smoothed prime ideals theorems under GRH, to appear in Math. Comp., 2015, DOI: \\url{https:\/\/doi.org\/10.1090\/mcom3039}.","DOI":"10.1090\/mcom3039"},{"key":"6","doi-asserted-by":"crossref","unstructured":"L. Greni\u00e9 and G. Molteni, Explicit versions of the prime ideal theorem for Dedekind zeta functions under GRH, to appear in Math. Comp., 2015, DOI: \\url{https:\/\/doi.org\/10.1090\/mcom3031}.","DOI":"10.1090\/mcom3031"},{"key":"7","doi-asserted-by":"publisher","first-page":"337","DOI":"10.1007\/BF01403312","article-title":"On the zeta function of number fields","volume":"12","author":"Lang, Serge","year":"1971","journal-title":"Invent. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0020-9910","issn-type":"print"},{"key":"8","doi-asserted-by":"crossref","unstructured":"J. E. Littlewood, On the zeros of the Riemann Zeta-function, Cambr. Phil. Soc. Proc. 22 (1924), 295\u2013318.","DOI":"10.1017\/S0305004100014225"},{"key":"9","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-0175-5","volume-title":"Complex analysis in one variable","author":"Narasimhan, Raghavan","year":"2001","ISBN":"https:\/\/id.crossref.org\/isbn\/0817641645","edition":"2"},{"issue":"5","key":"10","first-page":"89","article-title":"Contributions to the theory of the Riemann zeta-function","volume":"48","author":"Selberg, Atle","year":"1946","journal-title":"Arch. Math. Naturvid.","ISSN":"https:\/\/id.crossref.org\/issn\/0365-4524","issn-type":"print"},{"key":"11","doi-asserted-by":"publisher","first-page":"135","DOI":"10.1007\/BF01405166","article-title":"Some effective cases of the Brauer-Siegel theorem","volume":"23","author":"Stark, H. M.","year":"1974","journal-title":"Invent. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0020-9910","issn-type":"print"},{"key":"12","isbn-type":"print","volume-title":"The theory of the Riemann zeta-function","author":"Titchmarsh, E. C.","year":"1986","ISBN":"https:\/\/id.crossref.org\/isbn\/0198533691","edition":"2"},{"key":"13","doi-asserted-by":"crossref","unstructured":"T. S. Trudgian, An improved upper bound for the error in the zero-counting formulae for Dirichlet L-functions and Dedekind zeta-functions, Math. Comp. 84 (2015), 1439\u20131450. DOI: \\url{https:\/\/doi.org\/10.1090\/S0025-5718-2014-02898-6}, 2014.","DOI":"10.1090\/S0025-5718-2014-02898-6"},{"key":"14","doi-asserted-by":"publisher","first-page":"280","DOI":"10.1016\/j.jnt.2013.07.017","article-title":"An improved upper bound for the argument of the Riemann zeta-function on the critical line II","volume":"134","author":"Trudgian, Timothy S.","year":"2014","journal-title":"J. Number Theory","ISSN":"https:\/\/id.crossref.org\/issn\/0022-314X","issn-type":"print"}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/www.ams.org\/mcom\/2016-85-299\/S0025-5718-2015-03024-5\/S0025-5718-2015-03024-5.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/mcom\/2016-85-299\/S0025-5718-2015-03024-5\/S0025-5718-2015-03024-5.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T18:45:50Z","timestamp":1776797150000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/2016-85-299\/S0025-5718-2015-03024-5\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2015,10,9]]},"references-count":14,"journal-issue":{"issue":"299","published-print":{"date-parts":[[2016,5]]}},"alternative-id":["S0025-5718-2015-03024-5"],"URL":"https:\/\/doi.org\/10.1090\/mcom\/3024","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":[[2015,10,9]]}}}