{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,22]],"date-time":"2026-04-22T13:11:53Z","timestamp":1776863513593,"version":"3.51.2"},"reference-count":13,"publisher":"American Mathematical Society (AMS)","issue":"213","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>\n                    Strengthening work of Rosser, Schoenfeld, and McCurley, we establish explicit Chebyshev-type estimates in the prime number theorem for arithmetic progressions, for all moduli\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"k less-than-or-equal-to 72\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>k<\/mml:mi>\n                            <mml:mo>\n                              \u2264\n                              \n                            <\/mml:mo>\n                            <mml:mn>72<\/mml:mn>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">k \\le 72<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    and other small moduli.\n                  <\/p>","DOI":"10.1090\/s0025-5718-96-00669-2","type":"journal-article","created":{"date-parts":[[2002,7,26]],"date-time":"2002-07-26T18:14:44Z","timestamp":1027707284000},"page":"397-425","source":"Crossref","is-referenced-by-count":63,"title":["Primes in arithmetic progressions"],"prefix":"10.1090","volume":"65","author":[{"given":"Olivier","family":"Ramar\u00e9","sequence":"first","affiliation":[]},{"given":"Robert","family":"Rumely","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[1996]]},"reference":[{"key":"1","unstructured":"E. Landau, Handbuch der Lehre von der Verteilung der Primzahlen, with an appendix by P. Bateman, 3rd edition, Chelsea, New York, 1974."},{"key":"2","doi-asserted-by":"crossref","unstructured":"J. van de Lune, H. J. J. te Riele, and D. T. Winter,  On the zeros of the Riemann zeta function in the critical strip. IV, Math. Comp. 46 (1986), 667\u2013681.","DOI":"10.1090\/S0025-5718-1986-0829637-3"},{"key":"3","unstructured":"K. S. McCurley, Explicit estimates for functions of primes in arithmetic progressions, Ph.D. thesis, University of Illinois at Urbana-Champagne, 1981."},{"key":"4","doi-asserted-by":"crossref","unstructured":"K. S. McCurley, Explicit zero-free regions for Dirichlet \ud835\udc3f-functions, J. Number Theory  19 (1984), 7\u201332.","DOI":"10.1016\/0022-314X(84)90089-1"},{"key":"5","doi-asserted-by":"crossref","unstructured":"K. S. McCurley, Explicit estimates for the error term in the prime number theorem for arithmetic progressions, Math. Comp. 42 (1984), 265\u2013285.","DOI":"10.1090\/S0025-5718-1984-0726004-6"},{"key":"6","doi-asserted-by":"crossref","unstructured":"K. S. McCurley, Explicit estimates for \ud835\udf03(\ud835\udc4b;3,\ud835\udc59) and \ud835\udf13(\ud835\udc4b;3,\ud835\udc59), Math. Comp.  42 (1984), 287\u2013296.","DOI":"10.1090\/S0025-5718-1984-0726005-8"},{"key":"7","unstructured":"W. Press, B. Flannery, S. Teukolsky, and W. Vetterling, Numerical recipes, Cambridge Univ. Press, Cambridge, 1986."},{"key":"8","doi-asserted-by":"crossref","unstructured":"J. B. Rosser, Explicit bounds for some functions of prime numbers, Amer. J. Math. 63 (1941), 211\u2013232.","DOI":"10.2307\/2371291"},{"key":"9","doi-asserted-by":"crossref","unstructured":"J. B. Rosser and L. Schoenfeld, Sharper bounds for the Chebyshev functions \ud835\udf03(\ud835\udc4b) and \ud835\udf13(\ud835\udc4b), Math. Comp. 29 (1975), 243\u2013269.","DOI":"10.1090\/S0025-5718-1975-0457373-7"},{"key":"10","doi-asserted-by":"crossref","unstructured":"R. Rumely, Numerical computations concerning the ERH, Math. Comp. 62 (1993), 415\u2013440.","DOI":"10.1090\/S0025-5718-1993-1195435-0"},{"key":"11","doi-asserted-by":"crossref","unstructured":"L. Schoenfeld, Sharper bounds for the Chebyshev functions \ud835\udf03(\ud835\udc4b) and \ud835\udf13(\ud835\udc4b). II, Math. Comp. 30 (1976), 337\u2013360.","DOI":"10.1090\/S0025-5718-1976-0457374-X"},{"key":"12","unstructured":"S. B. Stechkin, Rational inequalities and zeros of the Riemann zeta-function, Trudy Mat. Inst. Steklov.  189 (1989), 110\u2013116. English transl. in Proc. Steklov Inst. Math. (189) (1990) 127\u2013134."},{"key":"13","doi-asserted-by":"crossref","unstructured":"R. Terras, A Miller algorithm for an incomplete Bessel function, J. Comput. Phys. 39 (1981), 233\u2013240.","DOI":"10.1016\/0021-9991(81)90147-9"}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/www.ams.org\/mcom\/1996-65-213\/S0025-5718-96-00669-2\/S0025-5718-96-00669-2.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/mcom\/1996-65-213\/S0025-5718-96-00669-2\/S0025-5718-96-00669-2.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,20]],"date-time":"2026-04-20T21:02:45Z","timestamp":1776718965000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/1996-65-213\/S0025-5718-96-00669-2\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1996]]},"references-count":13,"journal-issue":{"issue":"213","published-print":{"date-parts":[[1996,1]]}},"alternative-id":["S0025-5718-96-00669-2"],"URL":"https:\/\/doi.org\/10.1090\/s0025-5718-96-00669-2","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":[[1996]]}}}