{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,20]],"date-time":"2026-04-20T23:38:41Z","timestamp":1776728321833,"version":"3.51.2"},"reference-count":9,"publisher":"American Mathematical Society (AMS)","issue":"239","license":[{"start":{"date-parts":[[2002,11,21]],"date-time":"2002-11-21T00:00:00Z","timestamp":1037836800000},"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                    We extend a result of Ramar\u00e9 and Rumely, 1996, about the Chebyshev function\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"theta\">\n                        <mml:semantics>\n                          <mml:mi>\n                            \u03b8\n                            \n                          <\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">\\theta<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    in arithmetic progressions. We find a map\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"epsilon left-parenthesis x right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>\n                              \u03b5\n                              \n                            <\/mml:mi>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>x<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\varepsilon (x)<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    such that\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"bar theta left-parenthesis x semicolon k comma l right-parenthesis minus x slash phi left-parenthesis k right-parenthesis bar greater-than x epsilon left-parenthesis x right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mo stretchy=\"false\">\n                              \u2223\n                              \n                            <\/mml:mo>\n                            <mml:mi>\n                              \u03b8\n                              \n                            <\/mml:mi>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>x<\/mml:mi>\n                            <mml:mo>;<\/mml:mo>\n                            <mml:mi>k<\/mml:mi>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mi>l<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                            <mml:mo>\n                              \u2212\n                              \n                            <\/mml:mo>\n                            <mml:mi>x<\/mml:mi>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mo>\/<\/mml:mo>\n                            <\/mml:mrow>\n                            <mml:mi>\n                              \u03c6\n                              \n                            <\/mml:mi>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>k<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                            <mml:mo>\n                              \u2223\n                              \n                              &gt;\n                            <\/mml:mo>\n                            <mml:mi>x<\/mml:mi>\n                            <mml:mi>\n                              \u03b5\n                              \n                            <\/mml:mi>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>x<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\mid \\theta (x;k,l)-x\/\\varphi (k)\\mid &gt;x\\varepsilon (x)<\/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=\"epsilon left-parenthesis x right-parenthesis equals upper O left-parenthesis StartFraction 1 Over ln Superscript a Baseline x EndFraction right-parenthesis left-parenthesis for-all a greater-than 0 right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>\n                              \u03b5\n                              \n                            <\/mml:mi>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>x<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:mi>O<\/mml:mi>\n                            <mml:mrow>\n                              <mml:mo>(<\/mml:mo>\n                              <mml:mfrac>\n                                <mml:mn>1<\/mml:mn>\n                                <mml:mrow>\n                                  <mml:msup>\n                                    <mml:mi>ln<\/mml:mi>\n                                    <mml:mi>a<\/mml:mi>\n                                  <\/mml:msup>\n                                  <mml:mo>\n                                    \u2061\n                                    \n                                  <\/mml:mo>\n                                  <mml:mi>x<\/mml:mi>\n                                <\/mml:mrow>\n                              <\/mml:mfrac>\n                              <mml:mo>)<\/mml:mo>\n                            <\/mml:mrow>\n                            <mml:mspace width=\"1em\"\/>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mo stretchy=\"false\">(<\/mml:mo>\n                              <mml:mi mathvariant=\"normal\">\n                                \u2200\n                                \n                              <\/mml:mi>\n                              <mml:mi>a<\/mml:mi>\n                              <mml:mo>&gt;<\/mml:mo>\n                              <mml:mn>0<\/mml:mn>\n                              <mml:mo stretchy=\"false\">)<\/mml:mo>\n                            <\/mml:mrow>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\varepsilon (x)=O\\left (\\frac {1}{\\ln ^a x}\\right )\\quad {(\\forall a&gt;0)}<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    , whereas\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"epsilon left-parenthesis x right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>\n                              \u03b5\n                              \n                            <\/mml:mi>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>x<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\varepsilon (x)<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    is a constant. Now we are able to show that, for\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"x greater-than-or-slanted-equals 1531\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>x<\/mml:mi>\n                            <mml:mo>\n                              \u2a7e\n                              \n                            <\/mml:mo>\n                            <mml:mn>1531<\/mml:mn>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">x\\geqslant 1531<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    ,\n                    <disp-formula content-type=\"math\/mathml\">\n                      \\[\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"bar theta left-parenthesis x semicolon 3 comma l right-parenthesis minus x slash 2 bar greater-than 0.262 StartFraction x Over ln x EndFraction\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mo stretchy=\"false\">\n                              \u2223\n                              \n                            <\/mml:mo>\n                            <mml:mi>\n                              \u03b8\n                              \n                            <\/mml:mi>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>x<\/mml:mi>\n                            <mml:mo>;<\/mml:mo>\n                            <mml:mn>3<\/mml:mn>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mi>l<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                            <mml:mo>\n                              \u2212\n                              \n                            <\/mml:mo>\n                            <mml:mi>x<\/mml:mi>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mo>\/<\/mml:mo>\n                            <\/mml:mrow>\n                            <mml:mn>2<\/mml:mn>\n                            <mml:mo>\n                              \u2223\n                              \n                              &gt;\n                            <\/mml:mo>\n                            <mml:mn>0.262<\/mml:mn>\n                            <mml:mfrac>\n                              <mml:mi>x<\/mml:mi>\n                              <mml:mrow>\n                                <mml:mi>ln<\/mml:mi>\n                                <mml:mo>\n                                  \u2061\n                                  \n                                <\/mml:mo>\n                                <mml:mi>x<\/mml:mi>\n                              <\/mml:mrow>\n                            <\/mml:mfrac>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\mid \\theta (x;3,l)-x\/2\\mid &gt;0.262\\frac {x}{\\ln x}<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                      \\]\n                    <\/disp-formula>\n                    and, for\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"x greater-than-or-slanted-equals 151\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>x<\/mml:mi>\n                            <mml:mo>\n                              \u2a7e\n                              \n                            <\/mml:mo>\n                            <mml:mn>151<\/mml:mn>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">x\\geqslant 151<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    ,\n                    <disp-formula content-type=\"math\/mathml\">\n                      \\[\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"pi left-parenthesis x semicolon 3 comma l right-parenthesis greater-than StartFraction x Over 2 ln x EndFraction period\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>\n                              \u03c0\n                              \n                            <\/mml:mi>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>x<\/mml:mi>\n                            <mml:mo>;<\/mml:mo>\n                            <mml:mn>3<\/mml:mn>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mi>l<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                            <mml:mo>&gt;<\/mml:mo>\n                            <mml:mfrac>\n                              <mml:mi>x<\/mml:mi>\n                              <mml:mrow>\n                                <mml:mn>2<\/mml:mn>\n                                <mml:mi>ln<\/mml:mi>\n                                <mml:mo>\n                                  \u2061\n                                  \n                                <\/mml:mo>\n                                <mml:mi>x<\/mml:mi>\n                              <\/mml:mrow>\n                            <\/mml:mfrac>\n                            <mml:mo>.<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\pi (x;3,l)&gt;\\frac {x}{2\\ln x}.<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                      \\]\n                    <\/disp-formula>\n                  <\/p>","DOI":"10.1090\/s0025-5718-01-01351-5","type":"journal-article","created":{"date-parts":[[2002,9,20]],"date-time":"2002-09-20T15:46:54Z","timestamp":1032536814000},"page":"1137-1168","source":"Crossref","is-referenced-by-count":12,"title":["Estimates of \ud835\udf03(\ud835\udc65;\ud835\udc58,\ud835\udc59) for large values of \ud835\udc65"],"prefix":"10.1090","volume":"71","author":[{"given":"Pierre","family":"Dusart","sequence":"first","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2001,11,21]]},"reference":[{"issue":"1","key":"1","doi-asserted-by":"publisher","first-page":"7","DOI":"10.1016\/0022-314X(84)90089-1","article-title":"Explicit zero-free regions for Dirichlet \ud835\udc3f-functions","volume":"19","author":"McCurley, Kevin 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IV","volume":"46","author":"van de Lune, J.","year":"1986","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/www.ams.org\/mcom\/2002-71-239\/S0025-5718-01-01351-5\/S0025-5718-01-01351-5.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/mcom\/2002-71-239\/S0025-5718-01-01351-5\/S0025-5718-01-01351-5.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,20]],"date-time":"2026-04-20T23:00:31Z","timestamp":1776726031000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/2002-71-239\/S0025-5718-01-01351-5\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2001,11,21]]},"references-count":9,"journal-issue":{"issue":"239","published-print":{"date-parts":[[2002,7]]}},"alternative-id":["S0025-5718-01-01351-5"],"URL":"https:\/\/doi.org\/10.1090\/s0025-5718-01-01351-5","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":[[2001,11,21]]}}}