{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,27]],"date-time":"2026-03-27T05:19:40Z","timestamp":1774588780420,"version":"3.50.1"},"reference-count":18,"publisher":"American Mathematical Society (AMS)","issue":"254","license":[{"start":{"date-parts":[[2006,12,2]],"date-time":"2006-12-02T00:00:00Z","timestamp":1165017600000},"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><inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"normal upper Psi left-parenthesis x comma y right-parenthesis\">\n  <mml:semantics>\n    <mml:mrow>\n      <mml:mi mathvariant=\"normal\">\u03a8<\/mml:mi>\n      <mml:mo stretchy=\"false\">(<\/mml:mo>\n      <mml:mi>x<\/mml:mi>\n      <mml:mo>,<\/mml:mo>\n      <mml:mi>y<\/mml:mi>\n      <mml:mo stretchy=\"false\">)<\/mml:mo>\n    <\/mml:mrow>\n    <mml:annotation encoding=\"application\/x-tex\">\\Psi (x,y)<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula> denotes the number of positive integers <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"less-than-or-equal-to x\">\n  <mml:semantics>\n    <mml:mrow>\n      <mml:mo>\u2264<\/mml:mo>\n      <mml:mi>x<\/mml:mi>\n    <\/mml:mrow>\n    <mml:annotation encoding=\"application\/x-tex\">\\leq x<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula> and free of prime factors <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"greater-than y\">\n  <mml:semantics>\n    <mml:mrow>\n      <mml:mo>&gt;<\/mml:mo>\n      <mml:mi>y<\/mml:mi>\n    <\/mml:mrow>\n    <mml:annotation encoding=\"application\/x-tex\">&gt;y<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula>. Hildebrand and Tenenbaum gave a smooth approximation formula for <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"normal upper Psi left-parenthesis x comma y right-parenthesis\">\n  <mml:semantics>\n    <mml:mrow>\n      <mml:mi mathvariant=\"normal\">\u03a8<\/mml:mi>\n      <mml:mo stretchy=\"false\">(<\/mml:mo>\n      <mml:mi>x<\/mml:mi>\n      <mml:mo>,<\/mml:mo>\n      <mml:mi>y<\/mml:mi>\n      <mml:mo stretchy=\"false\">)<\/mml:mo>\n    <\/mml:mrow>\n    <mml:annotation encoding=\"application\/x-tex\">\\Psi (x,y)<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula> in the range <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"left-parenthesis log x right-parenthesis Superscript 1 plus epsilon Baseline greater-than y less-than-or-equal-to x\">\n  <mml:semantics>\n    <mml:mrow>\n      <mml:mo stretchy=\"false\">(<\/mml:mo>\n      <mml:mi>log<\/mml:mi>\n      <mml:mo>\u2061<\/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:mo>+<\/mml:mo>\n          <mml:mi>\u03f5<\/mml:mi>\n        <\/mml:mrow>\n      <\/mml:msup>\n      <mml:mo>&gt;<\/mml:mo>\n      <mml:mi>y<\/mml:mi>\n      <mml:mo>\u2264<\/mml:mo>\n      <mml:mi>x<\/mml:mi>\n    <\/mml:mrow>\n    <mml:annotation encoding=\"application\/x-tex\">(\\log x)^{1+\\epsilon }&gt; y \\leq x<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula>, where <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"epsilon\">\n  <mml:semantics>\n    <mml:mi>\u03f5<\/mml:mi>\n    <mml:annotation encoding=\"application\/x-tex\">\\epsilon<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula> is a fixed positive number <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"less-than-or-equal-to 1 slash 2\">\n  <mml:semantics>\n    <mml:mrow>\n      <mml:mo>\u2264<\/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:mn>2<\/mml:mn>\n    <\/mml:mrow>\n    <mml:annotation encoding=\"application\/x-tex\">\\leq 1\/2<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula>. In this paper, by modifying their approximation formula, we provide a fast algorithm to approximate <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"normal upper Psi left-parenthesis x comma y right-parenthesis\">\n  <mml:semantics>\n    <mml:mrow>\n      <mml:mi mathvariant=\"normal\">\u03a8<\/mml:mi>\n      <mml:mo stretchy=\"false\">(<\/mml:mo>\n      <mml:mi>x<\/mml:mi>\n      <mml:mo>,<\/mml:mo>\n      <mml:mi>y<\/mml:mi>\n      <mml:mo stretchy=\"false\">)<\/mml:mo>\n    <\/mml:mrow>\n    <mml:annotation encoding=\"application\/x-tex\">\\Psi (x,y)<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula>. The computational complexity of this algorithm is <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper O left-parenthesis StartRoot left-parenthesis log x right-parenthesis left-parenthesis log y right-parenthesis EndRoot right-parenthesis\">\n  <mml:semantics>\n    <mml:mrow>\n      <mml:mi>O<\/mml:mi>\n      <mml:mo stretchy=\"false\">(<\/mml:mo>\n      <mml:msqrt>\n        <mml:mo stretchy=\"false\">(<\/mml:mo>\n        <mml:mi>log<\/mml:mi>\n        <mml:mo>\u2061<\/mml:mo>\n        <mml:mi>x<\/mml:mi>\n        <mml:mo stretchy=\"false\">)<\/mml:mo>\n        <mml:mo stretchy=\"false\">(<\/mml:mo>\n        <mml:mi>log<\/mml:mi>\n        <mml:mo>\u2061<\/mml:mo>\n        <mml:mi>y<\/mml:mi>\n        <mml:mo stretchy=\"false\">)<\/mml:mo>\n      <\/mml:msqrt>\n      <mml:mo stretchy=\"false\">)<\/mml:mo>\n    <\/mml:mrow>\n    <mml:annotation encoding=\"application\/x-tex\">O(\\sqrt {(\\log x)(\\log y)})<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula>. We give numerical results which show that this algorithm provides accurate estimates for <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"normal upper Psi left-parenthesis x comma y right-parenthesis\">\n  <mml:semantics>\n    <mml:mrow>\n      <mml:mi mathvariant=\"normal\">\u03a8<\/mml:mi>\n      <mml:mo stretchy=\"false\">(<\/mml:mo>\n      <mml:mi>x<\/mml:mi>\n      <mml:mo>,<\/mml:mo>\n      <mml:mi>y<\/mml:mi>\n      <mml:mo stretchy=\"false\">)<\/mml:mo>\n    <\/mml:mrow>\n    <mml:annotation encoding=\"application\/x-tex\">\\Psi (x,y)<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula> and is faster than conventional methods such as algorithms exploiting Dickman\u2019s function.<\/p>","DOI":"10.1090\/s0025-5718-05-01798-9","type":"journal-article","created":{"date-parts":[[2006,2,15]],"date-time":"2006-02-15T16:05:20Z","timestamp":1140019520000},"page":"1015-1024","source":"Crossref","is-referenced-by-count":4,"title":["Approximating the number of integers without large prime factors"],"prefix":"10.1090","volume":"75","author":[{"given":"Koji","family":"Suzuki","sequence":"first","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2005,12,2]]},"reference":[{"issue":"246","key":"1","doi-asserted-by":"publisher","first-page":"1023","DOI":"10.1090\/S0025-5718-03-01501-1","article-title":"Prime sieves using binary quadratic forms","volume":"73","author":"Atkin, A. O. L.","year":"2004","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"216","key":"2","doi-asserted-by":"publisher","first-page":"1701","DOI":"10.1090\/S0025-5718-96-00775-2","article-title":"Asymptotic semismoothness probabilities","volume":"65","author":"Bach, Eric","year":"1996","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"key":"3","doi-asserted-by":"publisher","first-page":"128","DOI":"10.1007\/BFb0054856","article-title":"Bounding smooth integers (extended abstract)","author":"Bernstein, Daniel J.","year":"1998"},{"issue":"1","key":"4","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1016\/0022-314X(83)90002-1","article-title":"On a problem of Oppenheim concerning \u201cfactorisatio numerorum\u201d","volume":"17","author":"Canfield, E. R.","year":"1983","journal-title":"J. Number Theory","ISSN":"https:\/\/id.crossref.org\/issn\/0022-314X","issn-type":"print"},{"issue":"191","key":"5","doi-asserted-by":"publisher","first-page":"129","DOI":"10.2307\/2008795","article-title":"A differential delay equation arising from the sieve of Eratosthenes","volume":"55","author":"Cheer, A. Y.","year":"1990","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"key":"6","first-page":"50","article-title":"On the number of positive integers \u2264\ud835\udc65 and free of prime factors >\ud835\udc66","volume":"54","author":"de Bruijn, N. G.","year":"1951","journal-title":"Nederl. Acad. Wetensch. Proc. Ser. A."},{"key":"7","first-page":"25","article-title":"The asymptotic behaviour of a function occurring in the theory of primes","volume":"15","author":"de Bruijn, N. G.","year":"1951","journal-title":"J. Indian Math. Soc. (N.S.)","ISSN":"https:\/\/id.crossref.org\/issn\/0019-5839","issn-type":"print"},{"key":"8","unstructured":"K. Dickman, On the frequency of numbers containing prime factors of a certain relative magnitude, Arkiv F\u00f6r Matematik, Astromi Fysik. Band 22 A, no. 10, 1-14, 1930."},{"issue":"3","key":"9","doi-asserted-by":"publisher","first-page":"289","DOI":"10.1016\/0022-314X(86)90013-2","article-title":"On the number of positive integers \u2264\ud835\udc65 and free of prime factors >\ud835\udc66","volume":"22","author":"Hildebrand, Adolf","year":"1986","journal-title":"J. Number Theory","ISSN":"https:\/\/id.crossref.org\/issn\/0022-314X","issn-type":"print"},{"issue":"2","key":"10","doi-asserted-by":"publisher","first-page":"729","DOI":"10.2307\/2000551","article-title":"On the local behavior of \u03a8(\ud835\udc65,\ud835\udc66)","volume":"297","author":"Hildebrand, Adolf","year":"1986","journal-title":"Trans. Amer. Math. Soc.","ISSN":"https:\/\/id.crossref.org\/issn\/0002-9947","issn-type":"print"},{"issue":"1","key":"11","doi-asserted-by":"publisher","first-page":"265","DOI":"10.2307\/2000573","article-title":"On integers free of large prime factors","volume":"296","author":"Hildebrand, Adolf","year":"1986","journal-title":"Trans. Amer. Math. Soc.","ISSN":"https:\/\/id.crossref.org\/issn\/0002-9947","issn-type":"print"},{"issue":"2","key":"12","doi-asserted-by":"crossref","first-page":"411","DOI":"10.5802\/jtnb.101","article-title":"Integers without large prime factors","volume":"5","author":"Hildebrand, Adolf","year":"1993","journal-title":"J. Th\\'{e}or. Nombres Bordeaux","ISSN":"https:\/\/id.crossref.org\/issn\/1246-7405","issn-type":"print"},{"issue":"220","key":"13","doi-asserted-by":"publisher","first-page":"1729","DOI":"10.1090\/S0025-5718-97-00874-0","article-title":"Approximating the number of integers free of large prime factors","volume":"66","author":"Hunter, Simon","year":"1997","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"187","key":"14","doi-asserted-by":"publisher","first-page":"191","DOI":"10.2307\/2008355","article-title":"Numerical solution of some classical differential-difference equations","volume":"53","author":"Marsaglia, George","year":"1989","journal-title":"Math. 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