{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,20]],"date-time":"2026-04-20T22:39:30Z","timestamp":1776724770422,"version":"3.51.2"},"reference-count":8,"publisher":"American Mathematical Society (AMS)","issue":"224","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>\n                    Let\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"normal upper Lamda\">\n                        <mml:semantics>\n                          <mml:mi mathvariant=\"normal\">\n                            \u039b\n                            \n                          <\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">\\Lambda<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    denote the\n                    <italic>Von Mangoldt<\/italic>\n                    function and\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"psi left-parenthesis x right-parenthesis equals sigma-summation Underscript n less-than-or-equal-to x Endscripts normal upper Lamda left-parenthesis n right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>\n                              \u03c8\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:munder>\n                              <mml:mo>\n                                \u2211\n                                \n                              <\/mml:mo>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mi>n<\/mml:mi>\n                                <mml:mo>\n                                  \u2264\n                                  \n                                <\/mml:mo>\n                                <mml:mi>x<\/mml:mi>\n                              <\/mml:mrow>\n                            <\/mml:munder>\n                            <mml:mi mathvariant=\"normal\">\n                              \u039b\n                              \n                            <\/mml:mi>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>n<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\psi (x)=\\sum _{n \\leq x} \\Lambda (n)<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    . We describe an elementary method for computing isolated values of\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"psi left-parenthesis x right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>\n                              \u03c8\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\">\\psi (x)<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    . The complexity of the algorithm is\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper O left-parenthesis x Superscript 2 slash 3 Baseline left-parenthesis log log x right-parenthesis Superscript 1 slash 3 Baseline right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>O<\/mml:mi>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:msup>\n                              <mml:mi>x<\/mml:mi>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mn>2<\/mml:mn>\n                                <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                  <mml:mo>\/<\/mml:mo>\n                                <\/mml:mrow>\n                                <mml:mn>3<\/mml:mn>\n                              <\/mml:mrow>\n                            <\/mml:msup>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\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>x<\/mml:mi>\n                            <mml:msup>\n                              <mml:mo stretchy=\"false\">)<\/mml:mo>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mn>1<\/mml:mn>\n                                <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                  <mml:mo>\/<\/mml:mo>\n                                <\/mml:mrow>\n                                <mml:mn>3<\/mml:mn>\n                              <\/mml:mrow>\n                            <\/mml:msup>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">O(x^{2\/3}(\\log \\log x)^{1\/3})<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    time and\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper O left-parenthesis x Superscript 1 slash 3 Baseline left-parenthesis log log x right-parenthesis Superscript 2 slash 3 Baseline right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>O<\/mml:mi>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:msup>\n                              <mml:mi>x<\/mml:mi>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mn>1<\/mml:mn>\n                                <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                  <mml:mo>\/<\/mml:mo>\n                                <\/mml:mrow>\n                                <mml:mn>3<\/mml:mn>\n                              <\/mml:mrow>\n                            <\/mml:msup>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\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>x<\/mml:mi>\n                            <mml:msup>\n                              <mml:mo stretchy=\"false\">)<\/mml:mo>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mn>2<\/mml:mn>\n                                <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                  <mml:mo>\/<\/mml:mo>\n                                <\/mml:mrow>\n                                <mml:mn>3<\/mml:mn>\n                              <\/mml:mrow>\n                            <\/mml:msup>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">O(x^{1\/3}(\\log \\log x)^{2\/3})<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    space. A table of values of\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"psi left-parenthesis x right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>\n                              \u03c8\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\">\\psi (x)<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    for\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                    up to\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"10 Superscript 15\">\n                        <mml:semantics>\n                          <mml:msup>\n                            <mml:mn>10<\/mml:mn>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mn>15<\/mml:mn>\n                            <\/mml:mrow>\n                          <\/mml:msup>\n                          <mml:annotation encoding=\"application\/x-tex\">10^{15}<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    is included, and some times of computation are given.\n                  <\/p>","DOI":"10.1090\/s0025-5718-98-00977-6","type":"journal-article","created":{"date-parts":[[2002,7,26]],"date-time":"2002-07-26T18:14:28Z","timestamp":1027707268000},"page":"1691-1696","source":"Crossref","is-referenced-by-count":4,"title":["Computing \ud835\udf13(\ud835\udc65)"],"prefix":"10.1090","volume":"67","author":[{"given":"Marc","family":"Del\u00e9glise","sequence":"first","affiliation":[]},{"given":"Jo\u00ebl","family":"Rivat","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[1998]]},"reference":[{"key":"1","doi-asserted-by":"crossref","unstructured":"M. Del\u00e9glise and J. Rivat, Computing the summation of the M\u00f6bius function, Experimental Mathematics, 5 (1996), pp. 291\u2013295.","DOI":"10.1080\/10586458.1996.10504594"},{"issue":"213","key":"2","doi-asserted-by":"publisher","first-page":"235","DOI":"10.1090\/S0025-5718-96-00674-6","article-title":"Computing \ud835\udf0b(\ud835\udc65): the Meissel, Lehmer, Lagarias, Miller, Odlyzko method","volume":"65","author":"Del\u00e9glise, M.","year":"1996","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"6","key":"3","doi-asserted-by":"publisher","first-page":"1365","DOI":"10.4153\/CJM-1982-095-9","article-title":"Prime numbers in short intervals and a generalized Vaughan identity","volume":"34","author":"Heath-Brown, D. R.","year":"1982","journal-title":"Canadian J. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0008-414X","issn-type":"print"},{"key":"4","unstructured":"G. Hoheisel, Primzahlprobleme in der Analysis, Sitz. Preuss. Akad. Wiss., 33 (1930), pp. 3\u201311."},{"issue":"170","key":"5","doi-asserted-by":"publisher","first-page":"537","DOI":"10.2307\/2007973","article-title":"Computing \ud835\udf0b(\ud835\udc65): the Meissel-Lehmer method","volume":"44","author":"Lagarias, J. C.","year":"1985","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"2","key":"6","doi-asserted-by":"publisher","first-page":"173","DOI":"10.1016\/0196-6774(87)90037-X","article-title":"Computing \ud835\udf0b(\ud835\udc65): an analytic method","volume":"8","author":"Lagarias, J. C.","year":"1987","journal-title":"J. Algorithms","ISSN":"https:\/\/id.crossref.org\/issn\/0196-6774","issn-type":"print"},{"key":"7","doi-asserted-by":"publisher","first-page":"111","DOI":"10.4064\/aa-37-1-111-115","article-title":"An elementary method in prime number theory","volume":"37","author":"Vaughan, R. C.","year":"1980","journal-title":"Acta Arith.","ISSN":"https:\/\/id.crossref.org\/issn\/0065-1036","issn-type":"print"},{"key":"8","doi-asserted-by":"publisher","first-page":"82","DOI":"10.2307\/1989993","article-title":"Maximal orders in rational cyclic algebras of composite degree","volume":"46","author":"Perlis, Sam","year":"1939","journal-title":"Trans. Amer. Math. Soc.","ISSN":"https:\/\/id.crossref.org\/issn\/0002-9947","issn-type":"print"}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/www.ams.org\/mcom\/1998-67-224\/S0025-5718-98-00977-6\/S0025-5718-98-00977-6.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/mcom\/1998-67-224\/S0025-5718-98-00977-6\/S0025-5718-98-00977-6.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,20]],"date-time":"2026-04-20T21:52:01Z","timestamp":1776721921000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/1998-67-224\/S0025-5718-98-00977-6\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1998]]},"references-count":8,"journal-issue":{"issue":"224","published-print":{"date-parts":[[1998,10]]}},"alternative-id":["S0025-5718-98-00977-6"],"URL":"https:\/\/doi.org\/10.1090\/s0025-5718-98-00977-6","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]]}}}