{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,22]],"date-time":"2026-04-22T05:50:15Z","timestamp":1776837015174,"version":"3.51.2"},"reference-count":21,"publisher":"American Mathematical Society (AMS)","issue":"343","license":[{"start":{"date-parts":[[2024,4,21]],"date-time":"2024-04-21T00:00:00Z","timestamp":1713657600000},"content-version":"am","delay-in-days":366,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"funder":[{"DOI":"10.13039\/501100001665","name":"Agence Nationale de la Recherche","doi-asserted-by":"publisher","award":["ANR-20-CE91-0006"],"award-info":[{"award-number":["ANR-20-CE91-0006"]}],"id":[{"id":"10.13039\/501100001665","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001665","name":"Agence Nationale de la Recherche","doi-asserted-by":"publisher","award":["I 4845-N"],"award-info":[{"award-number":["I 4845-N"]}],"id":[{"id":"10.13039\/501100001665","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001665","name":"Agence Nationale de la Recherche","doi-asserted-by":"publisher","award":["5401-1"],"award-info":[{"award-number":["5401-1"]}],"id":[{"id":"10.13039\/501100001665","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100002428","name":"Austrian Science Fund","doi-asserted-by":"publisher","award":["ANR-20-CE91-0006"],"award-info":[{"award-number":["ANR-20-CE91-0006"]}],"id":[{"id":"10.13039\/501100002428","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100002428","name":"Austrian Science Fund","doi-asserted-by":"publisher","award":["I 4845-N"],"award-info":[{"award-number":["I 4845-N"]}],"id":[{"id":"10.13039\/501100002428","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100002428","name":"Austrian Science Fund","doi-asserted-by":"publisher","award":["5401-1"],"award-info":[{"award-number":["5401-1"]}],"id":[{"id":"10.13039\/501100002428","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001852","name":"Indo-French Centre for the Promotion of Advanced Research","doi-asserted-by":"publisher","award":["ANR-20-CE91-0006"],"award-info":[{"award-number":["ANR-20-CE91-0006"]}],"id":[{"id":"10.13039\/501100001852","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001852","name":"Indo-French Centre for the Promotion of Advanced Research","doi-asserted-by":"publisher","award":["I 4845-N"],"award-info":[{"award-number":["I 4845-N"]}],"id":[{"id":"10.13039\/501100001852","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001852","name":"Indo-French Centre for the Promotion of Advanced Research","doi-asserted-by":"publisher","award":["5401-1"],"award-info":[{"award-number":["5401-1"]}],"id":[{"id":"10.13039\/501100001852","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>\n                    For all\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\\ge 2<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    and for all invertible residue classes\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"a\">\n                        <mml:semantics>\n                          <mml:mi>a<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">a<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    modulo\u00a0\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                    , there exists a natural number that is congruent to\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"a\">\n                        <mml:semantics>\n                          <mml:mi>a<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">a<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    modulo\u00a0\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                    and that is the product of exactly three primes, all of which are below\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"left-parenthesis 10 Superscript 15 Baseline q right-parenthesis Superscript 5 slash 2\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\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:mi>q<\/mml:mi>\n                            <mml:msup>\n                              <mml:mo stretchy=\"false\">)<\/mml:mo>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mn>5<\/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:msup>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">(10^{15}q)^{5\/2}<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    .\n                  <\/p>","DOI":"10.1090\/mcom\/3853","type":"journal-article","created":{"date-parts":[[2023,4,5]],"date-time":"2023-04-05T09:15:38Z","timestamp":1680686138000},"page":"2381-2411","source":"Crossref","is-referenced-by-count":0,"title":["Explicit bounds for products of primes in AP"],"prefix":"10.1090","volume":"92","author":[{"given":"Ramachandran","family":"Balasubramanian","sequence":"first","affiliation":[]},{"given":"Olivier","family":"Ramar\u00e9","sequence":"additional","affiliation":[]},{"given":"Priyamvad","family":"Srivastav","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2023,4,21]]},"reference":[{"issue":"4","key":"1","doi-asserted-by":"publisher","first-page":"843","DOI":"10.1142\/S1793042123500422","article-title":"Product of three primes in large arithmetic progressions","volume":"19","author":"Balasubramanian, Ramachandran","year":"2023","journal-title":"Int. J. Number Theory","ISSN":"https:\/\/id.crossref.org\/issn\/1793-0421","issn-type":"print"},{"issue":"331","key":"2","doi-asserted-by":"publisher","first-page":"2281","DOI":"10.1090\/mcom\/3643","article-title":"Sharper bounds for the Chebyshev function \ud835\udf03(\ud835\udc65)","volume":"90","author":"Broadbent, Samuel","year":"2021","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"key":"3","isbn-type":"print","doi-asserted-by":"publisher","first-page":"58","DOI":"10.1134\/S0371968517010058","article-title":"Quotients and products of zero-density subsets of the set of positive integers","volume":"296","author":"Silleruelo, Kh.","year":"2017","ISBN":"https:\/\/id.crossref.org\/isbn\/5784601415","journal-title":"Tr. Mat. Inst. Steklova","ISSN":"https:\/\/id.crossref.org\/issn\/0371-9685","issn-type":"print"},{"key":"4","doi-asserted-by":"publisher","first-page":"51","DOI":"10.7169\/facm\/1229618741","article-title":"Explicit estimates for summatory functions linked to the M\u00f6bius \ud835\udf07-function","volume":"37","author":"Cohen, Henri","year":"2007","journal-title":"Funct. Approx. Comment. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0208-6573","issn-type":"print"},{"issue":"1","key":"5","doi-asserted-by":"publisher","first-page":"91","DOI":"10.5802\/jtnb.1194","article-title":"Explicit \ud835\udc3f\u00b2 bounds for the Riemann \ud835\udf01 function","volume":"34","author":"Dona, Daniele","year":"2022","journal-title":"J. Th\\'{e}or. Nombres Bordeaux","ISSN":"https:\/\/id.crossref.org\/issn\/1246-7405","issn-type":"print"},{"issue":"3","key":"6","doi-asserted-by":"publisher","first-page":"229","DOI":"10.1007\/BF01848086","article-title":"On the residues of products of prime numbers","volume":"18","author":"Erd\u0151s, P.","year":"1987","journal-title":"Period. Math. Hungar.","ISSN":"https:\/\/id.crossref.org\/issn\/0031-5303","issn-type":"print"},{"key":"7","doi-asserted-by":"publisher","first-page":"329","DOI":"10.1007\/BF01403187","article-title":"A large sieve density estimate near \ud835\udf0e=1","volume":"11","author":"Gallagher, P. X.","year":"1970","journal-title":"Invent. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0020-9910","issn-type":"print"},{"key":"8","doi-asserted-by":"publisher","first-page":"60","DOI":"10.1515\/crll.1972.255.60","article-title":"On the Brun-Titchmarsh theorem","volume":"255","author":"Hooley, Christopher","year":"1972","journal-title":"J. Reine Angew. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0075-4102","issn-type":"print"},{"issue":"3","key":"9","doi-asserted-by":"publisher","first-page":"729","DOI":"10.1093\/qmath\/hax001","article-title":"A lower bound for the least prime in an arithmetic progression","volume":"68","author":"Li, Junxian","year":"2017","journal-title":"Q. J. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0033-5606","issn-type":"print"},{"key":"10","doi-asserted-by":"publisher","first-page":"303","DOI":"10.4153\/CMB-1966-039-4","article-title":"The error term for the squarefree integers","volume":"9","author":"Moser, L.","year":"1966","journal-title":"Canad. Math. Bull.","ISSN":"https:\/\/id.crossref.org\/issn\/0008-4395","issn-type":"print"},{"issue":"328","key":"11","doi-asserted-by":"publisher","first-page":"907","DOI":"10.1090\/mcom\/3581","article-title":"The distribution of \ud835\udc58-free numbers","volume":"90","author":"Mossinghoff, Michael J.","year":"2021","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"key":"12","doi-asserted-by":"publisher","first-page":"306","DOI":"10.2969\/jmsj\/02620306","article-title":"On some improvements of the Brun-Titchmarsh theorem","volume":"26","author":"Motohashi, Yoichi","year":"1974","journal-title":"J. Math. Soc. Japan","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5645","issn-type":"print"},{"issue":"6","key":"13","doi-asserted-by":"publisher","first-page":"1925","DOI":"10.1090\/S0002-9939-2014-12199-X","article-title":"The smallest prime that splits completely in an abelian number field","volume":"142","author":"Pollack, Paul","year":"2014","journal-title":"Proc. Amer. Math. Soc.","ISSN":"https:\/\/id.crossref.org\/issn\/0002-9939","issn-type":"print"},{"issue":"293","key":"14","doi-asserted-by":"publisher","first-page":"1359","DOI":"10.1090\/S0025-5718-2014-02914-1","article-title":"Explicit estimates on several summatory functions involving the Moebius function","volume":"84","author":"Ramar\u00e9, Olivier","year":"2015","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"297","key":"15","doi-asserted-by":"publisher","first-page":"325","DOI":"10.1090\/mcom\/2991","article-title":"An explicit density estimate for Dirichlet \ud835\udc3f-series","volume":"85","author":"Ramar\u00e9, O.","year":"2016","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"4","key":"16","doi-asserted-by":"publisher","first-page":"747","DOI":"10.1142\/S1793042120500384","article-title":"Product of primes in arithmetic progressions","volume":"16","author":"Ramar\u00e9, Olivier","year":"2020","journal-title":"Int. J. Number Theory","ISSN":"https:\/\/id.crossref.org\/issn\/1793-0421","issn-type":"print"},{"issue":"1","key":"17","doi-asserted-by":"publisher","first-page":"219","DOI":"10.5802\/jtnb.1024","article-title":"Products of primes in arithmetic progressions: a footnote in parity breaking","volume":"30","author":"Ramar\u00e9, Olivier","year":"2018","journal-title":"J. Th\\'{e}or. Nombres Bordeaux","ISSN":"https:\/\/id.crossref.org\/issn\/1246-7405","issn-type":"print"},{"key":"18","isbn-type":"print","first-page":"153","article-title":"Explicit average orders: news and problems","author":"Ramar\u00e9, Olivier","year":"2019","ISBN":"https:\/\/id.crossref.org\/isbn\/9788386806447"},{"key":"19","series-title":"North-Holland Series in Applied Mathematics and Mechanics, Vol. 10","volume-title":"Probability theory","author":"R\u00e9nyi, A.","year":"1970"},{"key":"20","doi-asserted-by":"publisher","first-page":"243","DOI":"10.2307\/2005479","article-title":"Sharper bounds for the Chebyshev functions \ud835\udf03(\ud835\udc65) and \ud835\udf13(\ud835\udc65)","volume":"29","author":"Rosser, J. Barkley","year":"1975","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"279","key":"21","doi-asserted-by":"publisher","first-page":"1777","DOI":"10.1090\/S0025-5718-2012-02579-8","article-title":"The least inert prime in a real quadratic field","volume":"81","author":"Trevi\u00f1o, Enrique","year":"2012","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.ams.org\/mcom\/2023-92-343\/S0025-5718-2023-03853-4\/mcom3853_AM.pdf","content-type":"application\/pdf","content-version":"am","intended-application":"syndication"},{"URL":"https:\/\/www.ams.org\/mcom\/2023-92-343\/S0025-5718-2023-03853-4\/S0025-5718-2023-03853-4.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,22]],"date-time":"2026-04-22T05:01:16Z","timestamp":1776834076000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/2023-92-343\/S0025-5718-2023-03853-4\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,4,21]]},"references-count":21,"journal-issue":{"issue":"343","published-print":{"date-parts":[[2023,9]]}},"alternative-id":["S0025-5718-2023-03853-4"],"URL":"https:\/\/doi.org\/10.1090\/mcom\/3853","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":[[2023,4,21]]}}}