{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,22]],"date-time":"2026-04-22T15:56:24Z","timestamp":1776873384733,"version":"3.51.2"},"reference-count":21,"publisher":"American Mathematical Society (AMS)","issue":"331","license":[{"start":{"date-parts":[[2022,4,5]],"date-time":"2022-04-05T00:00:00Z","timestamp":1649116800000},"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                    For every modulus\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"q greater-than-or-equal-to 3\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>q<\/mml:mi>\n                            <mml:mo>\n                              \u2265\n                              \n                            <\/mml:mo>\n                            <mml:mn>3<\/mml:mn>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">q\\ge 3<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    , we define a family of subsets\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"script upper A\">\n                        <mml:semantics>\n                          <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                            <mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">A<\/mml:mi>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\mathcal {A}<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    of the multiplicative group\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"left-parenthesis double-struck upper Z slash q double-struck upper Z right-parenthesis Superscript times\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mi mathvariant=\"double-struck\">Z<\/mml:mi>\n                            <\/mml:mrow>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mo>\/<\/mml:mo>\n                            <\/mml:mrow>\n                            <mml:mi>q<\/mml:mi>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mi mathvariant=\"double-struck\">Z<\/mml:mi>\n                            <\/mml:mrow>\n                            <mml:msup>\n                              <mml:mo stretchy=\"false\">)<\/mml:mo>\n                              <mml:mo>\n                                \u00d7\n                                \n                              <\/mml:mo>\n                            <\/mml:msup>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">(\\mathbb {Z}\/q\\mathbb {Z})^\\times<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    for which the Euler product\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"product Underscript p plus q double-struck upper Z element-of script upper A Endscripts left-parenthesis 1 minus p Superscript negative s Baseline right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:munder>\n                              <mml:mo>\n                                \u220f\n                                \n                              <\/mml:mo>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mi>p<\/mml:mi>\n                                <mml:mo>+<\/mml:mo>\n                                <mml:mi>q<\/mml:mi>\n                                <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                  <mml:mi mathvariant=\"double-struck\">Z<\/mml:mi>\n                                <\/mml:mrow>\n                                <mml:mo>\n                                  \u2208\n                                  \n                                <\/mml:mo>\n                                <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                  <mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">A<\/mml:mi>\n                                <\/mml:mrow>\n                              <\/mml:mrow>\n                            <\/mml:munder>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mn>1<\/mml:mn>\n                            <mml:mo>\n                              \u2212\n                              \n                            <\/mml:mo>\n                            <mml:msup>\n                              <mml:mi>p<\/mml:mi>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mo>\n                                  \u2212\n                                  \n                                <\/mml:mo>\n                                <mml:mi>s<\/mml:mi>\n                              <\/mml:mrow>\n                            <\/mml:msup>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\prod _{p+q\\mathbb {Z}\\in \\mathcal {A}}(1-p^{-s})<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    can be computed with high numerical precision, where\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"s greater-than 1\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>s<\/mml:mi>\n                            <mml:mo>&gt;<\/mml:mo>\n                            <mml:mn>1<\/mml:mn>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">s&gt;1<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    is some given real number. We provide a Sage script to do so, and extend our result to compute Euler products\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"product Underscript p plus q double-struck upper Z element-of script upper A Endscripts upper F left-parenthesis 1 slash p Superscript s Baseline right-parenthesis slash upper H left-parenthesis 1 slash p Superscript s Baseline right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:munder>\n                              <mml:mo>\n                                \u220f\n                                \n                              <\/mml:mo>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mi>p<\/mml:mi>\n                                <mml:mo>+<\/mml:mo>\n                                <mml:mi>q<\/mml:mi>\n                                <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                  <mml:mi mathvariant=\"double-struck\">Z<\/mml:mi>\n                                <\/mml:mrow>\n                                <mml:mo>\n                                  \u2208\n                                  \n                                <\/mml:mo>\n                                <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                  <mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">A<\/mml:mi>\n                                <\/mml:mrow>\n                              <\/mml:mrow>\n                            <\/mml:munder>\n                            <mml:mi>F<\/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:msup>\n                              <mml:mi>p<\/mml:mi>\n                              <mml:mi>s<\/mml:mi>\n                            <\/mml:msup>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mo>\/<\/mml:mo>\n                            <\/mml:mrow>\n                            <mml:mi>H<\/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:msup>\n                              <mml:mi>p<\/mml:mi>\n                              <mml:mi>s<\/mml:mi>\n                            <\/mml:msup>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\prod _{p+q\\mathbb {Z}\\in \\mathcal {A}}F(1\/p^s)\/H(1\/p^s)<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    where\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper F\">\n                        <mml:semantics>\n                          <mml:mi>F<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">F<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    and\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper H\">\n                        <mml:semantics>\n                          <mml:mi>H<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">H<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    are polynomials with real coefficients, when this product converges absolutely. This enables us to give precise values of several Euler products occurring in number theory.\n                  <\/p>","DOI":"10.1090\/mcom\/3630","type":"journal-article","created":{"date-parts":[[2021,1,13]],"date-time":"2021-01-13T11:40:14Z","timestamp":1610538014000},"page":"2247-2265","source":"Crossref","is-referenced-by-count":5,"title":["Fast multi-precision computation of some Euler products"],"prefix":"10.1090","volume":"90","author":[{"given":"S.","family":"Ettahri","sequence":"first","affiliation":[]},{"given":"O.","family":"Ramar\u00e9","sequence":"additional","affiliation":[]},{"given":"L.","family":"Surel","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2021,4,5]]},"reference":[{"key":"1","unstructured":"H. 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