{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,22]],"date-time":"2026-04-22T07:56:24Z","timestamp":1776844584343,"version":"3.51.2"},"reference-count":26,"publisher":"American Mathematical Society (AMS)","issue":"300","license":[{"start":{"date-parts":[[2016,10,6]],"date-time":"2016-10-06T00:00:00Z","timestamp":1475712000000},"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":"

\n Let\n \n \n \n \n \n \u03c8\n \n <\/mml:mi>\n \n \n K<\/mml:mi>\n <\/mml:mrow>\n <\/mml:mrow>\n <\/mml:msub>\n \\psi _{\\mathbb {K}}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n be the Chebyshev function of a number field\n \n \n \n \n K<\/mml:mi>\n <\/mml:mrow>\n \\mathbb {K}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n . Let\n \n \n \n \n \n \n \u03c8\n \n <\/mml:mi>\n \n \n K<\/mml:mi>\n <\/mml:mrow>\n <\/mml:mrow>\n \n (<\/mml:mo>\n 1<\/mml:mn>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:msubsup>\n (<\/mml:mo>\n x<\/mml:mi>\n )<\/mml:mo>\n :=<\/mml:mo>\n \n \n \u222b\n \n <\/mml:mo>\n \n 0<\/mml:mn>\n <\/mml:mrow>\n \n x<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msubsup>\n \n \n \u03c8\n \n <\/mml:mi>\n \n \n K<\/mml:mi>\n <\/mml:mrow>\n <\/mml:mrow>\n <\/mml:msub>\n (<\/mml:mo>\n t<\/mml:mi>\n )<\/mml:mo>\n \n \n d<\/mml:mi>\n <\/mml:mrow>\n t<\/mml:mi>\n <\/mml:mrow>\n \\psi ^{(1)}_{\\mathbb {K}}(x):=\\int _{0}^{x}\\psi _{\\mathbb {K}}(t)\\,\\mathrm {d} t<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n and\n \n \n \n \n \n \n \u03c8\n \n <\/mml:mi>\n \n \n K<\/mml:mi>\n <\/mml:mrow>\n <\/mml:mrow>\n \n (<\/mml:mo>\n 2<\/mml:mn>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:msubsup>\n (<\/mml:mo>\n x<\/mml:mi>\n )<\/mml:mo>\n :=<\/mml:mo>\n 2<\/mml:mn>\n \n \n \u222b\n \n <\/mml:mo>\n \n 0<\/mml:mn>\n <\/mml:mrow>\n \n x<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msubsup>\n \n \n \u03c8\n \n <\/mml:mi>\n \n \n K<\/mml:mi>\n <\/mml:mrow>\n <\/mml:mrow>\n \n (<\/mml:mo>\n 1<\/mml:mn>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:msubsup>\n (<\/mml:mo>\n t<\/mml:mi>\n )<\/mml:mo>\n \n \n d<\/mml:mi>\n <\/mml:mrow>\n t<\/mml:mi>\n <\/mml:mrow>\n \\psi ^{(2)}_{\\mathbb {K}}(x):=2\\int _{0}^{x}\\psi ^{(1)}_{\\mathbb {K}}(t)\\,\\mathrm {d} t<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n . We prove under GRH (Generalized Riemann Hypothesis) explicit inequalities for the differences\n \n \n \n \n \n |<\/mml:mo>\n <\/mml:mrow>\n \n \n \u03c8\n \n <\/mml:mi>\n \n \n K<\/mml:mi>\n <\/mml:mrow>\n <\/mml:mrow>\n \n (<\/mml:mo>\n 1<\/mml:mn>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:msubsup>\n (<\/mml:mo>\n x<\/mml:mi>\n )<\/mml:mo>\n \n \u2212\n \n <\/mml:mo>\n \n \n \n x<\/mml:mi>\n 2<\/mml:mn>\n <\/mml:msup>\n 2<\/mml:mn>\n <\/mml:mfrac>\n <\/mml:mstyle>\n \n |<\/mml:mo>\n <\/mml:mrow>\n <\/mml:mrow>\n |\\psi ^{(1)}_{\\mathbb {K}}(x) - \\tfrac {x^2}{2}|<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n and\n \n \n \n \n \n |<\/mml:mo>\n <\/mml:mrow>\n \n \n \u03c8\n \n <\/mml:mi>\n \n \n K<\/mml:mi>\n <\/mml:mrow>\n <\/mml:mrow>\n \n (<\/mml:mo>\n 2<\/mml:mn>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:msubsup>\n (<\/mml:mo>\n x<\/mml:mi>\n )<\/mml:mo>\n \n \u2212\n \n <\/mml:mo>\n \n \n \n x<\/mml:mi>\n 3<\/mml:mn>\n <\/mml:msup>\n 3<\/mml:mn>\n <\/mml:mfrac>\n <\/mml:mstyle>\n \n |<\/mml:mo>\n <\/mml:mrow>\n <\/mml:mrow>\n |\\psi ^{(2)}_{\\mathbb {K}}(x) - \\tfrac {x^3}{3}|<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n . We deduce an efficient algorithm for the computation of the residue of the Dedekind zeta function and a bound on small-norm prime ideals.\n <\/p>","DOI":"10.1090\/mcom3039","type":"journal-article","created":{"date-parts":[[2015,2,11]],"date-time":"2015-02-11T13:10:22Z","timestamp":1423660222000},"page":"1875-1899","source":"Crossref","is-referenced-by-count":9,"title":["Explicit smoothed prime ideals theorems 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":[[2015,10,6]]},"reference":[{"issue":"191","key":"1","doi-asserted-by":"publisher","first-page":"355","DOI":"10.2307\/2008811","article-title":"Explicit bounds for primality testing and related problems","volume":"55","author":"Bach, Eric","year":"1990","journal-title":"Math. 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