{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,22]],"date-time":"2026-04-22T08:36:59Z","timestamp":1776847019475,"version":"3.51.2"},"reference-count":7,"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                    A practical method to compute the Riemann zeta function is presented. The method can compute\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"zeta left-parenthesis 1 slash 2 plus i t right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>\n                              \u03b6\n                              \n                            <\/mml:mi>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mn>1<\/mml:mn>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mo>\/<\/mml:mo>\n                            <\/mml:mrow>\n                            <mml:mn>2<\/mml:mn>\n                            <mml:mo>+<\/mml:mo>\n                            <mml:mi>i<\/mml:mi>\n                            <mml:mi>t<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\zeta (1\/2+it)<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    at any\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"left floor upper T Superscript 1 slash 4 Baseline right floor\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mo fence=\"false\" stretchy=\"false\">\n                              \u230a\n                              \n                            <\/mml:mo>\n                            <mml:msup>\n                              <mml:mi>T<\/mml:mi>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mn>1<\/mml:mn>\n                                <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                  <mml:mo>\/<\/mml:mo>\n                                <\/mml:mrow>\n                                <mml:mn>4<\/mml:mn>\n                              <\/mml:mrow>\n                            <\/mml:msup>\n                            <mml:mo fence=\"false\" stretchy=\"false\">\n                              \u230b\n                              \n                            <\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\lfloor T^{1\/4} \\rfloor<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    points 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 comma upper T plus upper T Superscript 1 slash 4 Baseline right-bracket\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mo stretchy=\"false\">[<\/mml:mo>\n                            <mml:mi>T<\/mml:mi>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mi>T<\/mml:mi>\n                            <mml:mo>+<\/mml:mo>\n                            <mml:msup>\n                              <mml:mi>T<\/mml:mi>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mn>1<\/mml:mn>\n                                <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                  <mml:mo>\/<\/mml:mo>\n                                <\/mml:mrow>\n                                <mml:mn>4<\/mml:mn>\n                              <\/mml:mrow>\n                            <\/mml:msup>\n                            <mml:mo stretchy=\"false\">]<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">[T,T+T^{1\/4}]<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    using an\n                    <italic>average<\/italic>\n                    time of\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper T Superscript 1 slash 4 plus o left-parenthesis 1 right-parenthesis\">\n                        <mml:semantics>\n                          <mml:msup>\n                            <mml:mi>T<\/mml:mi>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mn>1<\/mml:mn>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mo>\/<\/mml:mo>\n                              <\/mml:mrow>\n                              <mml:mn>4<\/mml:mn>\n                              <mml:mo>+<\/mml:mo>\n                              <mml:mi>o<\/mml:mi>\n                              <mml:mo stretchy=\"false\">(<\/mml:mo>\n                              <mml:mn>1<\/mml:mn>\n                              <mml:mo stretchy=\"false\">)<\/mml:mo>\n                            <\/mml:mrow>\n                          <\/mml:msup>\n                          <mml:annotation encoding=\"application\/x-tex\">T^{1\/4+o(1)}<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    per point. This is the same complexity as the Odlyzko-Sch\u00f6nhage algorithm over that interval. Although the method far from competes with the Odlyzko-Sch\u00f6nhage algorithm over intervals much longer than\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper T Superscript 1 slash 4\">\n                        <mml:semantics>\n                          <mml:msup>\n                            <mml:mi>T<\/mml:mi>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mn>1<\/mml:mn>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mo>\/<\/mml:mo>\n                              <\/mml:mrow>\n                              <mml:mn>4<\/mml:mn>\n                            <\/mml:mrow>\n                          <\/mml:msup>\n                          <mml:annotation encoding=\"application\/x-tex\">T^{1\/4}<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    , it still has the advantages of being elementary, simple to implement, it does not use the fast Fourier transform or require large amounts of storage space, and its error terms are easy to control. The method has been implemented, and results of timing experiments agree with its theoretical amortized complexity of\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper T Superscript 1 slash 4 plus o left-parenthesis 1 right-parenthesis\">\n                        <mml:semantics>\n                          <mml:msup>\n                            <mml:mi>T<\/mml:mi>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mn>1<\/mml:mn>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mo>\/<\/mml:mo>\n                              <\/mml:mrow>\n                              <mml:mn>4<\/mml:mn>\n                              <mml:mo>+<\/mml:mo>\n                              <mml:mi>o<\/mml:mi>\n                              <mml:mo stretchy=\"false\">(<\/mml:mo>\n                              <mml:mn>1<\/mml:mn>\n                              <mml:mo stretchy=\"false\">)<\/mml:mo>\n                            <\/mml:mrow>\n                          <\/mml:msup>\n                          <mml:annotation encoding=\"application\/x-tex\">T^{1\/4+o(1)}<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    .\n                  <\/p>","DOI":"10.1090\/s0025-5718-2011-02452-x","type":"journal-article","created":{"date-parts":[[2011,1,25]],"date-time":"2011-01-25T08:27:46Z","timestamp":1295944066000},"page":"1785-1796","source":"Crossref","is-referenced-by-count":5,"title":["An amortized-complexity method to compute the Riemann zeta function"],"prefix":"10.1090","volume":"80","author":[{"given":"Ghaith","family":"Hiary","sequence":"first","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2011,1,25]]},"reference":[{"key":"1","unstructured":"[Ga] W. Gabcke, Neue Herleitung und explicite Restabsch\u00e4tzung der Riemann-Siegel-Formel. Ph.D. Dissertation, G\u00f6ttingen, 1979."},{"key":"2","unstructured":"[G] X. Gourdon, The 10\u00b9\u00b3 first zeros of the Riemann zeta function and zero computation at very large heights. Available at: http:\/\/numbers.compuation.free.fr."},{"issue":"2","key":"3","doi-asserted-by":"publisher","first-page":"372","DOI":"10.1137\/0519027","article-title":"Bounds for the tails of sharp-cutoff filter kernels","volume":"19","author":"Logan, B. F.","year":"1988","journal-title":"SIAM J. Math. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1410","issn-type":"print"},{"key":"4","unstructured":"[O] A.M. Odlyzko, The 10\u00b2\u2070-th zero of the Riemann zeta function and 175 million of its neighbors. www.dtc.umn.edu\/\u223codlyzko"},{"issue":"2","key":"5","doi-asserted-by":"publisher","first-page":"797","DOI":"10.2307\/2000939","article-title":"Fast algorithms for multiple evaluations of the Riemann zeta function","volume":"309","author":"Odlyzko, A. M.","year":"1988","journal-title":"Trans. Amer. Math. Soc.","ISSN":"https:\/\/id.crossref.org\/issn\/0002-9947","issn-type":"print"},{"key":"6","isbn-type":"print","doi-asserted-by":"publisher","first-page":"425","DOI":"10.1017\/CBO9780511550492.015","article-title":"Computational methods and experiments in analytic number theory","author":"Rubinstein, Michael","year":"2005","ISBN":"https:\/\/id.crossref.org\/isbn\/9780521620581"},{"key":"7","unstructured":"[R1] M.O. Rubinstein home-page, www.math.uwaterloo.ca\/\u223cmrubinst."}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/www.ams.org\/mcom\/2011-80-275\/S0025-5718-2011-02452-X\/S0025-5718-2011-02452-X.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/mcom\/2011-80-275\/S0025-5718-2011-02452-X\/S0025-5718-2011-02452-X.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T16:53:21Z","timestamp":1776790401000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/2011-80-275\/S0025-5718-2011-02452-X\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2011,1,25]]},"references-count":7,"journal-issue":{"issue":"275","published-print":{"date-parts":[[2011,7]]}},"alternative-id":["S0025-5718-2011-02452-X"],"URL":"https:\/\/doi.org\/10.1090\/s0025-5718-2011-02452-x","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":[[2011,1,25]]}}}