{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,20]],"date-time":"2026-04-20T22:39:02Z","timestamp":1776724742737,"version":"3.51.2"},"reference-count":14,"publisher":"American Mathematical Society (AMS)","issue":"223","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>\n                    We show that the multiple zeta sum:\n                    <disp-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"zeta left-parenthesis s 1 comma s 2 comma period period period comma s Subscript d Baseline right-parenthesis equals sigma-summation Underscript n 1 greater-than n 2 greater-than period period period greater-than n Subscript d Baseline Endscripts StartFraction 1 Over n 1 Superscript s 1 Baseline n 2 Superscript s 2 Baseline period period period n Subscript d Superscript s Super Subscript d Superscript Baseline EndFraction comma\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>\n                              \u03b6\n                              \n                            <\/mml:mi>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:msub>\n                              <mml:mi>s<\/mml:mi>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mn>1<\/mml:mn>\n                              <\/mml:mrow>\n                            <\/mml:msub>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:msub>\n                              <mml:mi>s<\/mml:mi>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mn>2<\/mml:mn>\n                              <\/mml:mrow>\n                            <\/mml:msub>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mo>.<\/mml:mo>\n                            <mml:mo>.<\/mml:mo>\n                            <mml:mo>.<\/mml:mo>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:msub>\n                              <mml:mi>s<\/mml:mi>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mi>d<\/mml:mi>\n                              <\/mml:mrow>\n                            <\/mml:msub>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:munder>\n                              <mml:mo>\n                                \u2211\n                                \n                              <\/mml:mo>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:msub>\n                                  <mml:mi>n<\/mml:mi>\n                                  <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                    <mml:mn>1<\/mml:mn>\n                                  <\/mml:mrow>\n                                <\/mml:msub>\n                                <mml:mo>&gt;<\/mml:mo>\n                                <mml:msub>\n                                  <mml:mi>n<\/mml:mi>\n                                  <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                    <mml:mn>2<\/mml:mn>\n                                  <\/mml:mrow>\n                                <\/mml:msub>\n                                <mml:mo>&gt;<\/mml:mo>\n                                <mml:mo>.<\/mml:mo>\n                                <mml:mo>.<\/mml:mo>\n                                <mml:mo>.<\/mml:mo>\n                                <mml:mo>&gt;<\/mml:mo>\n                                <mml:msub>\n                                  <mml:mi>n<\/mml:mi>\n                                  <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                    <mml:mi>d<\/mml:mi>\n                                  <\/mml:mrow>\n                                <\/mml:msub>\n                              <\/mml:mrow>\n                            <\/mml:munder>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mfrac>\n                                  <mml:mn>1<\/mml:mn>\n                                  <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                    <mml:msubsup>\n                                      <mml:mi>n<\/mml:mi>\n                                      <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                        <mml:mn>1<\/mml:mn>\n                                      <\/mml:mrow>\n                                      <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                        <mml:msub>\n                                          <mml:mi>s<\/mml:mi>\n                                          <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                            <mml:mn>1<\/mml:mn>\n                                          <\/mml:mrow>\n                                        <\/mml:msub>\n                                      <\/mml:mrow>\n                                    <\/mml:msubsup>\n                                    <mml:msubsup>\n                                      <mml:mi>n<\/mml:mi>\n                                      <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                        <mml:mn>2<\/mml:mn>\n                                      <\/mml:mrow>\n                                      <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                        <mml:msub>\n                                          <mml:mi>s<\/mml:mi>\n                                          <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                            <mml:mn>2<\/mml:mn>\n                                          <\/mml:mrow>\n                                        <\/mml:msub>\n                                      <\/mml:mrow>\n                                    <\/mml:msubsup>\n                                    <mml:mo>.<\/mml:mo>\n                                    <mml:mo>.<\/mml:mo>\n                                    <mml:mo>.<\/mml:mo>\n                                    <mml:msubsup>\n                                      <mml:mi>n<\/mml:mi>\n                                      <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                        <mml:mi>d<\/mml:mi>\n                                      <\/mml:mrow>\n                                      <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                        <mml:msub>\n                                          <mml:mi>s<\/mml:mi>\n                                          <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                            <mml:mi>d<\/mml:mi>\n                                          <\/mml:mrow>\n                                        <\/mml:msub>\n                                      <\/mml:mrow>\n                                    <\/mml:msubsup>\n                                  <\/mml:mrow>\n                                <\/mml:mfrac>\n                              <\/mml:mrow>\n                            <\/mml:mrow>\n                            <mml:mo>,<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\begin{equation*}\\zeta (s_{1}, s_{2}, ..., s_{d}) = \\sum _{n_{1} &gt; n_{2} &gt; ... &gt; n_{d}} {{\\frac {1 }{{n_{1}^{s_{1}} n_{2}^{s_{2}} ... n_{d}^{s_{d}}}}}},\\end{equation*}<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/disp-formula>\n                    for positive integers\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"s Subscript i\">\n                        <mml:semantics>\n                          <mml:msub>\n                            <mml:mi>s<\/mml:mi>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mi>i<\/mml:mi>\n                            <\/mml:mrow>\n                          <\/mml:msub>\n                          <mml:annotation encoding=\"application\/x-tex\">s_{i}<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    with\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"s 1 greater-than 1\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:msub>\n                              <mml:mi>s<\/mml:mi>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mn>1<\/mml:mn>\n                              <\/mml:mrow>\n                            <\/mml:msub>\n                            <mml:mo>&gt;<\/mml:mo>\n                            <mml:mn>1<\/mml:mn>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">s_{1}&gt;1<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    , can always be written as a finite sum of products of rapidly convergent series. Perhaps surprisingly, one may develop fast summation algorithms of such efficiency that the overall complexity can be brought down essentially to that of\n                    <italic>one<\/italic>\n                    -dimensional summation. In particular, for any dimension\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"d\">\n                        <mml:semantics>\n                          <mml:mi>d<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">d<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    one may resolve\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper D\">\n                        <mml:semantics>\n                          <mml:mi>D<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">D<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    good digits of\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"zeta\">\n                        <mml:semantics>\n                          <mml:mi>\n                            \u03b6\n                            \n                          <\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">\\zeta<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    in\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper O left-parenthesis upper D log upper D slash log log upper D right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>O<\/mml:mi>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>D<\/mml:mi>\n                            <mml:mi>log<\/mml:mi>\n                            <mml:mo>\n                              \u2061\n                              \n                            <\/mml:mo>\n                            <mml:mi>D<\/mml:mi>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mo>\/<\/mml:mo>\n                            <\/mml:mrow>\n                            <mml:mi>log<\/mml:mi>\n                            <mml:mo>\n                              \u2061\n                              \n                            <\/mml:mo>\n                            <mml:mi>log<\/mml:mi>\n                            <mml:mo>\n                              \u2061\n                              \n                            <\/mml:mo>\n                            <mml:mi>D<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">O(D \\log D \/ \\log \\log D)<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    arithmetic operations, with the implied big-\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper O\">\n                        <mml:semantics>\n                          <mml:mi>O<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">O<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    constant depending only on the set\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"StartSet s 1 comma period period period comma s Subscript d Baseline EndSet\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mo fence=\"false\" stretchy=\"false\">{<\/mml:mo>\n                            <mml:msub>\n                              <mml:mi>s<\/mml:mi>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mn>1<\/mml:mn>\n                              <\/mml:mrow>\n                            <\/mml:msub>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mo>.<\/mml:mo>\n                            <mml:mo>.<\/mml:mo>\n                            <mml:mo>.<\/mml:mo>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:msub>\n                              <mml:mi>s<\/mml:mi>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mi>d<\/mml:mi>\n                              <\/mml:mrow>\n                            <\/mml:msub>\n                            <mml:mo fence=\"false\" stretchy=\"false\">}<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\{s_{1},...,s_{d}\\}<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    .\n                  <\/p>","DOI":"10.1090\/s0025-5718-98-00950-8","type":"journal-article","created":{"date-parts":[[2002,7,26]],"date-time":"2002-07-26T18:14:28Z","timestamp":1027707268000},"page":"1163-1172","source":"Crossref","is-referenced-by-count":16,"title":["Fast evaluation of multiple zeta 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Math.","ISSN":"https:\/\/id.crossref.org\/issn\/1058-6458","issn-type":"print"},{"issue":"4","key":"12","first-page":"69","article-title":"Fast computation of the Riemann zeta function \ud835\udf01(\ud835\udc60) for integer values of \ud835\udc60","volume":"31","author":"Karatsuba, E. A.","year":"1995","journal-title":"Problemy Peredachi Informatsii","ISSN":"https:\/\/id.crossref.org\/issn\/0555-2923","issn-type":"print"},{"issue":"2","key":"13","doi-asserted-by":"publisher","first-page":"113","DOI":"10.1006\/jnth.1994.1058","article-title":"Triple sums and the Riemann zeta function","volume":"48","author":"Markett, C.","year":"1994","journal-title":"J. Number Theory","ISSN":"https:\/\/id.crossref.org\/issn\/0022-314X","issn-type":"print"},{"key":"14","unstructured":"D. Zagier, \u201cValues of zeta functions and their applications,\u201d preprint, Max-Planck Institut, 1994."}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/www.ams.org\/mcom\/1998-67-223\/S0025-5718-98-00950-8\/S0025-5718-98-00950-8.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/mcom\/1998-67-223\/S0025-5718-98-00950-8\/S0025-5718-98-00950-8.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,20]],"date-time":"2026-04-20T21:48:23Z","timestamp":1776721703000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/1998-67-223\/S0025-5718-98-00950-8\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1998]]},"references-count":14,"journal-issue":{"issue":"223","published-print":{"date-parts":[[1998,7]]}},"alternative-id":["S0025-5718-98-00950-8"],"URL":"https:\/\/doi.org\/10.1090\/s0025-5718-98-00950-8","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":[[1998]]}}}