{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,22]],"date-time":"2026-04-22T07:48:56Z","timestamp":1776844136323,"version":"3.51.2"},"reference-count":23,"publisher":"American Mathematical Society (AMS)","issue":"216","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>\n                    We call an integer\n                    <italic>semismooth<\/italic>\n                    with respect to\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"y\">\n                        <mml:semantics>\n                          <mml:mi>y<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">y<\/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\">\n                        <mml:semantics>\n                          <mml:mi>z<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">z<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    if each of its prime factors is\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"less-than-or-equal-to y\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mo>\n                              \u2264\n                              \n                            <\/mml:mo>\n                            <mml:mi>y<\/mml:mi>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\le y<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    , and all but one are\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"less-than-or-equal-to z\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mo>\n                              \u2264\n                              \n                            <\/mml:mo>\n                            <mml:mi>z<\/mml:mi>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\le z<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    . Such numbers are useful in various factoring algorithms, including the quadratic sieve. Let\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper G left-parenthesis alpha comma beta right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>G<\/mml:mi>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>\n                              \u03b1\n                              \n                            <\/mml:mi>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mi>\n                              \u03b2\n                              \n                            <\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">G(\\alpha ,\\beta )<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    be the asymptotic probability that a random integer\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"n\">\n                        <mml:semantics>\n                          <mml:mi>n<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">n<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    is semismooth with respect to\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"n Superscript beta\">\n                        <mml:semantics>\n                          <mml:msup>\n                            <mml:mi>n<\/mml:mi>\n                            <mml:mi>\n                              \u03b2\n                              \n                            <\/mml:mi>\n                          <\/mml:msup>\n                          <mml:annotation encoding=\"application\/x-tex\">n^\\beta<\/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=\"n Superscript alpha\">\n                        <mml:semantics>\n                          <mml:msup>\n                            <mml:mi>n<\/mml:mi>\n                            <mml:mi>\n                              \u03b1\n                              \n                            <\/mml:mi>\n                          <\/mml:msup>\n                          <mml:annotation encoding=\"application\/x-tex\">n^\\alpha<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    . We present new recurrence relations for\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper G\">\n                        <mml:semantics>\n                          <mml:mi>G<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">G<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    and related functions. We then give numerical methods for computing\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper G\">\n                        <mml:semantics>\n                          <mml:mi>G<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">G<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    , tables of\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper G\">\n                        <mml:semantics>\n                          <mml:mi>G<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">G<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    , and estimates for the error incurred by this asymptotic approximation.\n                  <\/p>","DOI":"10.1090\/s0025-5718-96-00775-2","type":"journal-article","created":{"date-parts":[[2002,7,26]],"date-time":"2002-07-26T18:14:28Z","timestamp":1027707268000},"page":"1701-1715","source":"Crossref","is-referenced-by-count":21,"title":["Asymptotic semismoothness probabilities"],"prefix":"10.1090","volume":"65","author":[{"given":"Eric","family":"Bach","sequence":"first","affiliation":[]},{"given":"Ren\u00e9","family":"Peralta","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[1996]]},"reference":[{"issue":"2","key":"1","doi-asserted-by":"publisher","first-page":"179","DOI":"10.1137\/0217012","article-title":"How to generate factored random numbers","volume":"17","author":"Bach, Eric","year":"1988","journal-title":"SIAM J. Comput.","ISSN":"https:\/\/id.crossref.org\/issn\/0097-5397","issn-type":"print"},{"key":"2","isbn-type":"print","first-page":"48","article-title":"Exact analysis of a priority queue algorithm for random variate generation","author":"Bach, Eric","year":"1994","ISBN":"https:\/\/id.crossref.org\/isbn\/0898713293"},{"key":"3","first-page":"623","article-title":"Annihilator ideals and representation iteration for abstract rings","volume":"5","author":"Everett, C. J., Jr.","year":"1939","journal-title":"Duke Math. J.","ISSN":"https:\/\/id.crossref.org\/issn\/0012-7094","issn-type":"print"},{"key":"4","first-page":"623","article-title":"Annihilator ideals and representation iteration for abstract rings","volume":"5","author":"Everett, C. J., Jr.","year":"1939","journal-title":"Duke Math. J.","ISSN":"https:\/\/id.crossref.org\/issn\/0012-7094","issn-type":"print"},{"key":"5","doi-asserted-by":"publisher","first-page":"197","DOI":"10.2307\/2005263","article-title":"A probabilistic approach to a differential-difference equation arising in analytic number theory","volume":"27","author":"Chamayou, Jean-Marie-Fran\u00e7ois","year":"1973","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"191","key":"6","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. 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Amer. Math. Soc.","ISSN":"https:\/\/id.crossref.org\/issn\/0002-9947","issn-type":"print"},{"key":"10","isbn-type":"print","doi-asserted-by":"publisher","first-page":"355","DOI":"10.1007\/3-540-46885-4_35","article-title":"Factoring by electronic mail","author":"Lenstra, Arjen K.","year":"1990","ISBN":"https:\/\/id.crossref.org\/isbn\/3540534334"},{"issue":"2","key":"11","doi-asserted-by":"publisher","first-page":"258","DOI":"10.1112\/S0025579300012481","article-title":"Integers free of large prime factors and the Riemann hypothesis","volume":"31","author":"Hildebrand, Adolf","year":"1984","journal-title":"Mathematika","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5793","issn-type":"print"},{"key":"12","unstructured":"R. Lambert, Computational aspects of discrete logarithms, Ph.D. thesis, University of Waterloo, 1996."},{"key":"13","doi-asserted-by":"publisher","first-page":"417","DOI":"10.2307\/2004437","article-title":"On the numerical solution of a differential-difference equation arising in analytic number theory","volume":"23","author":"van de Lune, J.","year":"1969","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. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"key":"15","unstructured":"P. L. Montgomery, An FFT extension of the elliptic curve method of factorization, Ph.D. thesis, University of California - Los Angeles, 1992."},{"issue":"190","key":"16","doi-asserted-by":"publisher","first-page":"839","DOI":"10.2307\/2008514","article-title":"An FFT extension to the \ud835\udc43-1 factoring algorithm","volume":"54","author":"Montgomery, Peter L.","year":"1990","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"key":"17","series-title":"Memoirs of the American Mathematical Society, No. 106","volume-title":"Numbers with small prime factors, and the least $k$th power non-residue","author":"Norton, Karl K.","year":"1971"},{"key":"18","unstructured":"N. Patterson, Letter to Eric Bach, November 1988."},{"key":"19","isbn-type":"print","doi-asserted-by":"publisher","first-page":"169","DOI":"10.1007\/3-540-39757-4_17","article-title":"The quadratic sieve factoring algorithm","author":"Pomerance, Carl","year":"1985","ISBN":"https:\/\/id.crossref.org\/isbn\/3540160760"},{"key":"20","doi-asserted-by":"publisher","first-page":"783","DOI":"10.2307\/2371336","article-title":"Ring homomorphisms which are also lattice homomorphisms","volume":"61","author":"Ward, Morgan","year":"1939","journal-title":"Amer. J. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0002-9327","issn-type":"print"},{"issue":"167","key":"21","doi-asserted-by":"publisher","first-page":"289","DOI":"10.2307\/2007414","article-title":"A Monte Carlo factoring algorithm with linear storage","volume":"43","author":"Schnorr, C.-P.","year":"1984","journal-title":"Math. 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