{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,12]],"date-time":"2026-06-12T18:00:02Z","timestamp":1781287202454,"version":"3.54.1"},"reference-count":12,"publisher":"American Mathematical Society (AMS)","issue":"361","license":[{"start":{"date-parts":[[2026,7,17]],"date-time":"2026-07-17T00:00:00Z","timestamp":1784246400000},"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 prove 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 q\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>x<\/mml:mi>\n                            <mml:mo>&gt;<\/mml:mo>\n                            <mml:mi>q<\/mml:mi>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">x &gt; q<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    (\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"x\">\n                        <mml:semantics>\n                          <mml:mi>x<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">x<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    real), uniformly in\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 2\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>q<\/mml:mi>\n                            <mml:mo>\n                              \u2265\n                              \n                            <\/mml:mo>\n                            <mml:mn>2<\/mml:mn>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">q \\geq 2<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    ,\n                    <disp-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"StartFraction phi left-parenthesis q right-parenthesis Over q EndFraction StartAbsoluteValue sigma-summation Underscript StartLayout 1st Row  j less-than-or-equal-to x 2nd Row  left-parenthesis j comma q right-parenthesis equals 1 EndLayout Endscripts StartFraction mu left-parenthesis j right-parenthesis Over j EndFraction EndAbsoluteValue log left-parenthesis x slash q right-parenthesis less-than-or-equal-to 0.3055 period\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mfrac>\n                              <mml:mrow>\n                                <mml:mi>\n                                  \u03c6\n                                  \n                                <\/mml:mi>\n                                <mml:mo stretchy=\"false\">(<\/mml:mo>\n                                <mml:mi>q<\/mml:mi>\n                                <mml:mo stretchy=\"false\">)<\/mml:mo>\n                              <\/mml:mrow>\n                              <mml:mi>q<\/mml:mi>\n                            <\/mml:mfrac>\n                            <mml:mstyle scriptlevel=\"0\">\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mo maxsize=\"2.047em\" minsize=\"2.047em\">|<\/mml:mo>\n                              <\/mml:mrow>\n                            <\/mml:mstyle>\n                            <mml:munder>\n                              <mml:mo>\n                                \u2211\n                                \n                              <\/mml:mo>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mstyle scriptlevel=\"1\">\n                                  <mml:mtable rowspacing=\"0.1em\" columnspacing=\"0em\" displaystyle=\"false\">\n                                    <mml:mtr>\n                                      <mml:mtd>\n                                        <mml:mi>j<\/mml:mi>\n                                        <mml:mo>\n                                          \u2264\n                                          \n                                        <\/mml:mo>\n                                        <mml:mi>x<\/mml:mi>\n                                      <\/mml:mtd>\n                                    <\/mml:mtr>\n                                    <mml:mtr>\n                                      <mml:mtd>\n                                        <mml:mo stretchy=\"false\">(<\/mml:mo>\n                                        <mml:mi>j<\/mml:mi>\n                                        <mml:mo>,<\/mml:mo>\n                                        <mml:mi>q<\/mml:mi>\n                                        <mml:mo stretchy=\"false\">)<\/mml:mo>\n                                        <mml:mo>=<\/mml:mo>\n                                        <mml:mn>1<\/mml:mn>\n                                      <\/mml:mtd>\n                                    <\/mml:mtr>\n                                  <\/mml:mtable>\n                                <\/mml:mstyle>\n                              <\/mml:mrow>\n                            <\/mml:munder>\n                            <mml:mfrac>\n                              <mml:mrow>\n                                <mml:mi>\n                                  \u03bc\n                                  \n                                <\/mml:mi>\n                                <mml:mo stretchy=\"false\">(<\/mml:mo>\n                                <mml:mi>j<\/mml:mi>\n                                <mml:mo stretchy=\"false\">)<\/mml:mo>\n                              <\/mml:mrow>\n                              <mml:mi>j<\/mml:mi>\n                            <\/mml:mfrac>\n                            <mml:mstyle scriptlevel=\"0\">\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mo maxsize=\"2.047em\" minsize=\"2.047em\">|<\/mml:mo>\n                              <\/mml:mrow>\n                            <\/mml:mstyle>\n                            <mml:mi>log<\/mml:mi>\n                            <mml:mo>\n                              \u2061\n                              \n                            <\/mml:mo>\n                            <mml:mo stretchy=\"false\">(<\/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>q<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                            <mml:mo>\n                              \u2264\n                              \n                            <\/mml:mo>\n                            <mml:mn>0.3055.<\/mml:mn>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\begin{equation*} \\frac {\\varphi (q)}{q}\\bigg |\\sum _{\\substack {j \\leq x \\\\ (j,q) = 1}} \\frac {\\mu (j)}{j}\\bigg | \\log (x\/q) \\leq 0.3055. \\end{equation*}<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/disp-formula>\n                    The constant\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"0.3055\">\n                        <mml:semantics>\n                          <mml:mn>0.3055<\/mml:mn>\n                          <mml:annotation encoding=\"application\/x-tex\">0.3055<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    is optimal up to the third decimal place. This answers a question of Ramar\u00e9 [Math. Comp. 84 (2015), pp. 1359\u20131387]. Sharper results are obtained for larger integers\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                    . Similar results are obtained for the sums\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"sigma-summation Underscript StartLayout 1st Row  j less-than-or-equal-to x 2nd Row  left-parenthesis j comma q right-parenthesis equals 1 EndLayout Endscripts mu left-parenthesis j right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mstyle displaystyle=\"true\" scriptlevel=\"0\">\n                            <mml:munder>\n                              <mml:mo>\n                                \u2211\n                                \n                              <\/mml:mo>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mstyle scriptlevel=\"1\">\n                                  <mml:mtable rowspacing=\"0.1em\" columnspacing=\"0em\" displaystyle=\"false\">\n                                    <mml:mtr>\n                                      <mml:mtd>\n                                        <mml:mi>j<\/mml:mi>\n                                        <mml:mo>\n                                          \u2264\n                                          \n                                        <\/mml:mo>\n                                        <mml:mi>x<\/mml:mi>\n                                      <\/mml:mtd>\n                                    <\/mml:mtr>\n                                    <mml:mtr>\n                                      <mml:mtd>\n                                        <mml:mo stretchy=\"false\">(<\/mml:mo>\n                                        <mml:mi>j<\/mml:mi>\n                                        <mml:mo>,<\/mml:mo>\n                                        <mml:mi>q<\/mml:mi>\n                                        <mml:mo stretchy=\"false\">)<\/mml:mo>\n                                        <mml:mo>=<\/mml:mo>\n                                        <mml:mn>1<\/mml:mn>\n                                      <\/mml:mtd>\n                                    <\/mml:mtr>\n                                  <\/mml:mtable>\n                                <\/mml:mstyle>\n                              <\/mml:mrow>\n                            <\/mml:munder>\n                            <mml:mi>\n                              \u03bc\n                              \n                            <\/mml:mi>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>j<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mstyle>\n                          <mml:annotation encoding=\"application\/x-tex\">\\displaystyle \\sum _{\\substack {j \\leq x \\\\ (j,q) = 1}} \\mu (j)<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    .\n                  <\/p>","DOI":"10.1090\/mcom\/4128","type":"journal-article","created":{"date-parts":[[2025,7,17]],"date-time":"2025-07-17T16:03:12Z","timestamp":1752768192000},"page":"2515-2538","source":"Crossref","is-referenced-by-count":0,"title":["Sharp estimates for the M\u00f6bius function with coprimality restrictions"],"prefix":"10.1090","volume":"95","author":[{"given":"Andr\u00e9","family":"de Camargo","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"14","published-online":{"date-parts":[[2025,7,17]]},"reference":[{"key":"1","unstructured":"O. 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Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"2","key":"12","doi-asserted-by":"publisher","first-page":"343","DOI":"10.1017\/S0004972709000884","article-title":"A remark on partial sums involving the M\u00f6bius function","volume":"81","author":"Tao, Terence","year":"2010","journal-title":"Bull. Aust. Math. Soc.","ISSN":"https:\/\/id.crossref.org\/issn\/0004-9727","issn-type":"print"}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.ams.org\/mcom\/2026-95-361\/S0025-5718-2025-04128-0\/S0025-5718-2025-04128-0.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,6,12]],"date-time":"2026-06-12T17:54:08Z","timestamp":1781286848000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/2026-95-361\/S0025-5718-2025-04128-0\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,7,17]]},"references-count":12,"journal-issue":{"issue":"361","published-print":{"date-parts":[[2026,9]]}},"alternative-id":["S0025-5718-2025-04128-0"],"URL":"https:\/\/doi.org\/10.1090\/mcom\/4128","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":[[2025,7,17]]}}}