{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T15:35:29Z","timestamp":1776785729479,"version":"3.51.2"},"reference-count":12,"publisher":"American Mathematical Society (AMS)","issue":"254","license":[{"start":{"date-parts":[[2006,12,20]],"date-time":"2006-12-20T00:00:00Z","timestamp":1166572800000},"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                    The three main methods used in diophantine analysis of\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"q\">\n                        <mml:semantics>\n                          <mml:mi>q<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">q<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    -series are combined to obtain new upper bounds for irrationality measures of the values of the\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"q\">\n                        <mml:semantics>\n                          <mml:mi>q<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">q<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    -logarithm function\n                    <disp-formula content-type=\"math\/mathml\">\n                      \\[\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"ln Subscript q Baseline left-parenthesis 1 minus z right-parenthesis equals sigma-summation Underscript nu equals 1 Overscript normal infinity Endscripts StartFraction z Superscript nu Baseline q Superscript nu Baseline Over 1 minus q Superscript nu Baseline EndFraction comma StartAbsoluteValue z EndAbsoluteValue less-than-or-slanted-equals 1 comma\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:msub>\n                              <mml:mi>ln<\/mml:mi>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n     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              <mml:mi>\n                                  \u03bd\n                                  \n                                <\/mml:mi>\n                                <mml:mo>=<\/mml:mo>\n                                <mml:mn>1<\/mml:mn>\n                              <\/mml:mrow>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mi mathvariant=\"normal\">\n                                  \u221e\n                                  \n                                <\/mml:mi>\n                              <\/mml:mrow>\n                            <\/mml:munderover>\n                            <mml:mfrac>\n                              <mml:mrow>\n                                <mml:msup>\n                                  <mml:mi>z<\/mml:mi>\n                                  <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                    <mml:mi>\n                                      \u03bd\n                                      \n                                    <\/mml:mi>\n                                  <\/mml:mrow>\n                                <\/mml:msup>\n                                <mml:msup>\n                                  <mml:mi>q<\/mml:mi>\n                                  <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                    <mml:mi>\n                                      \u03bd\n                                      \n                                    <\/mml:mi>\n                                  <\/mml:mrow>\n                                <\/mml:msup>\n                              <\/mml:mrow>\n                              <mml:mrow>\n                                <mml:mn>1<\/mml:mn>\n                                <mml:mo>\n                                  \u2212\n                                  \n                                <\/mml:mo>\n                                <mml:msup>\n                                  <mml:mi>q<\/mml:mi>\n                                  <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                    <mml:mi>\n                                      \u03bd\n                                      \n                                    <\/mml:mi>\n                                  <\/mml:mrow>\n                                <\/mml:msup>\n                              <\/mml:mrow>\n                            <\/mml:mfrac>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mspace width=\"2em\"\/>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mo stretchy=\"false\">|<\/mml:mo>\n                            <\/mml:mrow>\n                            <mml:mi>z<\/mml:mi>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mo stretchy=\"false\">|<\/mml:mo>\n                            <\/mml:mrow>\n                            <mml:mo>\n                              \u2a7d\n                              \n                            <\/mml:mo>\n                            <mml:mn>1<\/mml:mn>\n                            <mml:mo>,<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\ln _{q}(1-z)=\\sum _{\\nu =1}^{\\infty }\\frac {z^{\\nu }q^{\\nu }}{1-q^{\\nu }}, \\qquad |z|\\leqslant 1,<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                      \\]\n                    <\/disp-formula>\n                    when\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"p equals 1 slash q element-of double-struck upper Z minus StartSet 0 comma plus-or-minus 1 EndSet\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>p<\/mml:mi>\n                            <mml:mo>=<\/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:mi>q<\/mml:mi>\n                            <mml:mo>\n                              \u2208\n                              \n                            <\/mml:mo>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mi mathvariant=\"double-struck\">Z<\/mml:mi>\n                            <\/mml:mrow>\n                            <mml:mo class=\"MJX-variant\">\n                              \u2216\n                              \n                            <\/mml:mo>\n                            <mml:mo fence=\"false\" stretchy=\"false\">{<\/mml:mo>\n                            <mml:mn>0<\/mml:mn>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mo>\n                              \u00b1\n                              \n                            <\/mml:mo>\n                            <mml:mn>1<\/mml:mn>\n                            <mml:mo fence=\"false\" stretchy=\"false\">}<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">p=1\/q\\in \\mathbb {Z}\\setminus \\{0,\\pm 1\\}<\/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=\"z element-of double-struck upper Q\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>z<\/mml:mi>\n                            <mml:mo>\n                              \u2208\n                              \n                            <\/mml:mo>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mi mathvariant=\"double-struck\">Q<\/mml:mi>\n                            <\/mml:mrow>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">z\\in \\mathbb {Q}<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    .\n                  <\/p>","DOI":"10.1090\/s0025-5718-05-01812-0","type":"journal-article","created":{"date-parts":[[2006,2,15]],"date-time":"2006-02-15T11:05:20Z","timestamp":1140001520000},"page":"879-889","source":"Crossref","is-referenced-by-count":12,"title":["New irrationality measures for 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