{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,11]],"date-time":"2026-03-11T12:49:07Z","timestamp":1773233347868,"version":"3.50.1"},"reference-count":52,"publisher":"American Mathematical Society (AMS)","issue":"287","license":[{"start":{"date-parts":[[2014,8,20]],"date-time":"2014-08-20T00:00:00Z","timestamp":1408492800000},"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>Let <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"phi\">\n  <mml:semantics>\n    <mml:mi>\u03d5<\/mml:mi>\n    <mml:annotation encoding=\"application\/x-tex\">\\phi<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula> denote Euler\u2019s phi function. For a fixed odd prime <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> we investigate the first and second order terms of the asymptotic series expansion for the number of <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"n less-than-or-slanted-equals x\">\n  <mml:semantics>\n    <mml:mrow>\n      <mml:mi>n<\/mml:mi>\n      <mml:mo>\u2a7d<\/mml:mo>\n      <mml:mi>x<\/mml:mi>\n    <\/mml:mrow>\n    <mml:annotation encoding=\"application\/x-tex\">n\\leqslant x<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula> such that <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"q does-not-divide phi left-parenthesis n right-parenthesis\">\n  <mml:semantics>\n    <mml:mrow>\n      <mml:mi>q<\/mml:mi>\n      <mml:mo>\u2224<\/mml:mo>\n      <mml:mi>\u03d5<\/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\">q\\nmid \\phi (n)<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula>. Part of the analysis involves a careful study of the Euler-Kronecker constants for cyclotomic fields. In particular, we show that the Hardy-Littlewood conjecture about counts of prime <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"k\">\n  <mml:semantics>\n    <mml:mi>k<\/mml:mi>\n    <mml:annotation encoding=\"application\/x-tex\">k<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula>-tuples and a conjecture of Ihara about the distribution of these Euler-Kronecker constants cannot be both true.<\/p>","DOI":"10.1090\/s0025-5718-2013-02749-4","type":"journal-article","created":{"date-parts":[[2013,8,20]],"date-time":"2013-08-20T13:29:27Z","timestamp":1377005367000},"page":"1447-1476","source":"Crossref","is-referenced-by-count":22,"title":["Values of the Euler \ud835\udf19-function not divisible by a given odd prime, and the distribution of Euler-Kronecker constants for cyclotomic fields"],"prefix":"10.1090","volume":"83","author":[{"given":"Kevin","family":"Ford","sequence":"first","affiliation":[]},{"given":"Florian","family":"Luca","sequence":"additional","affiliation":[]},{"given":"Pieter","family":"Moree","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2013,8,20]]},"reference":[{"key":"1","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4614-3810-6","volume-title":"Ramanujan's lost notebook. Part III","author":"Andrews, George E.","year":"2012","ISBN":"https:\/\/id.crossref.org\/isbn\/9781461438090"},{"issue":"1","key":"2","doi-asserted-by":"publisher","first-page":"35","DOI":"10.1134\/S0001434610010050","article-title":"The Euler-Kronecker constant","volume":"87","author":"Badzyan, A. I.","year":"2010","journal-title":"Mat. Zametki","ISSN":"https:\/\/id.crossref.org\/issn\/0025-567X","issn-type":"print"},{"key":"3","first-page":"Art. B42c, 63","article-title":"Ramanujan\u2019s unpublished manuscript on the partition and tau functions with proofs and commentary","volume":"42","author":"Berndt, Bruce C.","year":"1999","journal-title":"S\\'{e}m. Lothar. Combin."},{"issue":"3","key":"4","doi-asserted-by":"publisher","first-page":"969","DOI":"10.4007\/annals.2006.163.969","article-title":"Classical and modular approaches to exponential Diophantine equations. I. Fibonacci and Lucas perfect powers","volume":"163","author":"Bugeaud, Yann","year":"2006","journal-title":"Ann. of Math. (2)","ISSN":"https:\/\/id.crossref.org\/issn\/0003-486X","issn-type":"print"},{"key":"5","series-title":"Graduate Texts in Mathematics","isbn-type":"print","volume-title":"Multiplicative number theory","volume":"74","author":"Davenport, Harold","year":"2000","ISBN":"https:\/\/id.crossref.org\/isbn\/0387950974","edition":"3"},{"key":"6","doi-asserted-by":"publisher","first-page":"105","DOI":"10.4064\/aa-19-2-105-146","article-title":"Sur des formules de Atle Selberg","volume":"19","author":"Delange, Hubert","year":"1971","journal-title":"Acta Arith.","ISSN":"https:\/\/id.crossref.org\/issn\/0065-1036","issn-type":"print"},{"key":"7","doi-asserted-by":"publisher","first-page":"171","DOI":"10.1515\/crll.1984.351.171","article-title":"On the analogue of the formula of Chowla and Selberg for real quadratic fields","volume":"351","author":"Deninger, Christopher","year":"1984","journal-title":"J. Reine Angew. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0075-4102","issn-type":"print"},{"key":"8","unstructured":"L.E. Dickson, A new extension of Dirichlet\u2019s theorem on prime numbers, Messenger of Math., 33 (1904), 155\u2013161."},{"key":"9","first-page":"59","article-title":"A conjecture in prime number theory","author":"Elliott, P. D. T. A.","year":"1970"},{"key":"10","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1017\/S144678870002632X","article-title":"On a problem of G. Golomb","volume":"2","author":"Erd\u0151s, P.","year":"1961","journal-title":"J. Austral. Math. Soc.","ISSN":"https:\/\/id.crossref.org\/issn\/0263-6115","issn-type":"print"},{"key":"11","unstructured":"FFTW Fast Fourier Transform C Library, available at http:\/\/www.fftw.org\/."},{"issue":"196","key":"12","doi-asserted-by":"publisher","first-page":"839","DOI":"10.2307\/2938721","article-title":"The prime factors of Wendt\u2019s binomial circulant determinant","volume":"57","author":"Fee, Greg","year":"1991","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"228","key":"13","doi-asserted-by":"publisher","first-page":"1739","DOI":"10.1090\/S0025-5718-99-01117-5","article-title":"Prime clusters and Cunningham chains","volume":"68","author":"Forbes, Tony","year":"1999","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"key":"14","unstructured":"T. Forbes, Prime \ud835\udc58-tuples, http:\/\/anthony.d.forbes.googlepages.com\/ktuplets.htm"},{"issue":"2","key":"15","doi-asserted-by":"publisher","first-page":"321","DOI":"10.1007\/BF01231189","article-title":"On the size of the first factor of the class number of a cyclotomic field","volume":"100","author":"Granville, A.","year":"1990","journal-title":"Invent. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0020-9910","issn-type":"print"},{"issue":"2","key":"16","doi-asserted-by":"publisher","first-page":"367","DOI":"10.4007\/annals.2008.168.367","article-title":"The distribution of integers with a divisor in a given interval","volume":"168","author":"Ford, Kevin","year":"2008","journal-title":"Ann. of Math. (2)","ISSN":"https:\/\/id.crossref.org\/issn\/0003-486X","issn-type":"print"},{"key":"17","series-title":"Cambridge Studies in Advanced Mathematics","isbn-type":"print","volume-title":"Algebraic number theory","volume":"27","author":"Fr\u00f6hlich, A.","year":"1993","ISBN":"https:\/\/id.crossref.org\/isbn\/0521438349"},{"key":"18","series-title":"London Mathematical Society Monographs, No. 4","volume-title":"Sieve methods","author":"Halberstam, H.","year":"1974"},{"key":"19","doi-asserted-by":"crossref","unstructured":"G.H. Hardy and J.E. Littlewood, Some problems of a \u201cPartitio Numerorum\u201d: III On the expression of a number as a sum of primes, Acta Math. 44 (1922), 1\u201370.","DOI":"10.1007\/BF02403921"},{"issue":"4","key":"20","first-page":"493","article-title":"Euler\u2019s constants for the Selberg and the Dedekind zeta functions","volume":"11","author":"Hashimoto, Yasufumi","year":"2004","journal-title":"Bull. Belg. Math. Soc. Simon Stevin","ISSN":"https:\/\/id.crossref.org\/issn\/1370-1444","issn-type":"print"},{"key":"21","first-page":"123","article-title":"On the incompatibility of two conjectures concerning primes","author":"Hensley, Douglas","year":"1973"},{"key":"22","doi-asserted-by":"publisher","first-page":"407","DOI":"10.1007\/978-0-8176-4532-8_5","article-title":"On the Euler-Kronecker constants of global fields and primes with small norms","author":"Ihara, Yasutaka","year":"2006"},{"key":"23","first-page":"79","article-title":"The Euler-Kronecker invariants in various families of global fields","author":"Ihara, Yasutaka","year":"2010"},{"issue":"3","key":"24","doi-asserted-by":"publisher","first-page":"253","DOI":"10.4064\/aa137-3-6","article-title":"On the logarithmic derivatives of Dirichlet \ud835\udc3f-functions at \ud835\udc60=1","volume":"137","author":"Ihara, Yasutaka","year":"2009","journal-title":"Acta Arith.","ISSN":"https:\/\/id.crossref.org\/issn\/0065-1036","issn-type":"print"},{"key":"25","unstructured":"E. Landau, \u00dcber die Einteilung der positiven ganzen Zahlen in vier Klassen nach der Mindestzahl der zu ihrer additiven Zusammensetzung erforderlichen Quadrate, Arch. der Math. und Phys. (3) 13 (1908), 305\u2013312. (See also his Collected Papers.)"},{"issue":"1","key":"26","doi-asserted-by":"publisher","first-page":"86","DOI":"10.2307\/2370180","article-title":"Losung des Lehmer\u2019schen Problems","volume":"31","author":"Landau, Edmund","year":"1909","journal-title":"Amer. J. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0002-9327","issn-type":"print"},{"key":"27","unstructured":"E. Landau, Handbuch der Lehre von der Verteilung der Primzahlen, 3rd ed., Chelsea, New York, 1953."},{"issue":"1","key":"28","doi-asserted-by":"publisher","first-page":"37","DOI":"10.1016\/j.jnt.2006.12.015","article-title":"A note on Mertens\u2019 formula for arithmetic progressions","volume":"127","author":"Languasco, A.","year":"2007","journal-title":"J. Number Theory","ISSN":"https:\/\/id.crossref.org\/issn\/0022-314X","issn-type":"print"},{"key":"29","isbn-type":"print","doi-asserted-by":"publisher","first-page":"17","DOI":"10.7169\/facm\/1269437065","article-title":"On the constant in the Mertens product for arithmetic progressions. I. Identities","volume":"42","author":"Languasco, Alessandro","year":"2010","ISBN":"https:\/\/id.crossref.org\/isbn\/9788323220824","journal-title":"Funct. Approx. Comment. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0208-6573","issn-type":"print"},{"issue":"265","key":"30","doi-asserted-by":"publisher","first-page":"315","DOI":"10.1090\/S0025-5718-08-02148-0","article-title":"On the constant in the Mertens product for arithmetic progressions. II. Numerical values","volume":"78","author":"Languasco, A.","year":"2009","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"3","key":"31","doi-asserted-by":"publisher","first-page":"279","DOI":"10.1080\/10586458.2010.10390624","article-title":"Computing the Mertens and Meissel-Mertens constants for sums over arithmetic progressions","volume":"19","author":"Languasco, Alessandro","year":"2010","journal-title":"Experiment. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/1058-6458","issn-type":"print"},{"issue":"4","key":"32","doi-asserted-by":"publisher","first-page":"333","DOI":"10.4064\/aa130-4-3","article-title":"Generalised Mertens and Brauer-Siegel theorems","volume":"130","author":"Lebacque, Philippe","year":"2007","journal-title":"Acta Arith.","ISSN":"https:\/\/id.crossref.org\/issn\/0065-1036","issn-type":"print"},{"issue":"6","key":"33","doi-asserted-by":"publisher","first-page":"125","DOI":"10.1070\/IM2000v064n06ABEH000314","article-title":"An explicit lower bound for a homogeneous rational linear form in logarithms of algebraic numbers. II","volume":"64","author":"Matveev, E. M.","year":"2000","journal-title":"Izv. Ross. Akad. Nauk Ser. Mat.","ISSN":"https:\/\/id.crossref.org\/issn\/1607-0046","issn-type":"print"},{"issue":"165","key":"34","doi-asserted-by":"publisher","first-page":"265","DOI":"10.2307\/2007579","article-title":"Explicit estimates for the error term in the prime number theorem for arithmetic progressions","volume":"42","author":"McCurley, Kevin S.","year":"1984","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"key":"35","doi-asserted-by":"publisher","first-page":"119","DOI":"10.1112\/S0025579300004708","article-title":"The large sieve","volume":"20","author":"Montgomery, H. L.","year":"1973","journal-title":"Mathematika","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5793","issn-type":"print"},{"key":"36","series-title":"Cambridge Studies in Advanced Mathematics","isbn-type":"print","volume-title":"Multiplicative number theory. I. Classical theory","volume":"97","author":"Montgomery, Hugh L.","year":"2007","ISBN":"https:\/\/id.crossref.org\/isbn\/9780521849036"},{"issue":"3","key":"37","doi-asserted-by":"publisher","first-page":"317","DOI":"10.1007\/s11139-004-0142-4","article-title":"On some claims in Ramanujan\u2019s \u2018unpublished\u2019 manuscript on the partition and tau functions","volume":"8","author":"Moree, Pieter","year":"2004","journal-title":"Ramanujan J.","ISSN":"https:\/\/id.crossref.org\/issn\/1382-4090","issn-type":"print"},{"issue":"245","key":"38","doi-asserted-by":"publisher","first-page":"425","DOI":"10.1090\/S0025-5718-03-01536-9","article-title":"Chebyshev\u2019s bias for composite numbers with restricted prime divisors","volume":"73","author":"Moree, Pieter","year":"2004","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"key":"39","unstructured":"P. Moree, Values of the Euler phi function not divisible by a prescribed odd prime, math.NT\/0611509, 2006, unpublished preprint."},{"issue":"3","key":"40","first-page":"73","article-title":"Counting numbers in multiplicative sets: Landau versus Ramanujan","volume":"21","author":"Moree, Pieter","year":"2011","journal-title":"Math. Newsl.","ISSN":"https:\/\/id.crossref.org\/issn\/0971-1694","issn-type":"print"},{"issue":"2","key":"41","first-page":"239","article-title":"The Euler-Kronecker constant of a cyclotomic field","volume":"35","author":"Murty, V. Kumar","year":"2011","journal-title":"Ann. Sci. Math. Qu\\'{e}bec","ISSN":"https:\/\/id.crossref.org\/issn\/0707-9109","issn-type":"print"},{"key":"42","isbn-type":"print","volume-title":"Elementary and analytic theory of algebraic numbers","author":"Narkiewicz, W\u0142adys\u0142aw","year":"1990","ISBN":"https:\/\/id.crossref.org\/isbn\/3540512500","edition":"2"},{"key":"43","unstructured":"OEIS Foundation (2011), The On-Line Encyclopedia of Integer Sequences, http:\/\/oeis.org\/."},{"issue":"1","key":"44","first-page":"1","article-title":"A generalization of Mertens\u2019 theorem","volume":"14","author":"Rosen, Michael","year":"1999","journal-title":"J. Ramanujan Math. Soc.","ISSN":"https:\/\/id.crossref.org\/issn\/0970-1249","issn-type":"print"},{"key":"45","first-page":"64","article-title":"Approximate formulas for some functions of prime numbers","volume":"6","author":"Rosser, J. Barkley","year":"1962","journal-title":"Illinois J. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0019-2082","issn-type":"print"},{"key":"46","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"},{"key":"47","doi-asserted-by":"publisher","first-page":"75","DOI":"10.2307\/2003407","article-title":"The second-order term in the asymptotic expansion of \ud835\udc35(\ud835\udc65)","volume":"18","author":"Shanks, Daniel","year":"1964","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"1","key":"48","doi-asserted-by":"publisher","first-page":"166","DOI":"10.1007\/s11512-005-0001-6","article-title":"Values of the Euler phi function not divisible by a given odd prime","volume":"44","author":"Spearman, Blair K.","year":"2006","journal-title":"Ark. Mat.","ISSN":"https:\/\/id.crossref.org\/issn\/0004-2080","issn-type":"print"},{"key":"49","series-title":"Cambridge Studies in Advanced Mathematics","isbn-type":"print","volume-title":"Introduction to analytic and probabilistic number theory","volume":"46","author":"Tenenbaum, G\u00e9rald","year":"1995","ISBN":"https:\/\/id.crossref.org\/isbn\/0521412617"},{"key":"50","doi-asserted-by":"publisher","first-page":"453","DOI":"10.1007\/978-0-8176-4532-8_6","article-title":"Asymptotic behaviour of the Euler-Kronecker constant","author":"Tsfasman, M. A.","year":"2006"},{"key":"51","unstructured":"C.J. de la Vall\u00e9e-Poussin, Recherches analytiques sur la th\u00e9orie des nombres premiers I, Ann. Soc. Sci. Bruxelles 20 (1896), 183\u2013256."},{"key":"52","series-title":"Graduate Texts in Mathematics","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4684-0133-2","volume-title":"Introduction to cyclotomic fields","volume":"83","author":"Washington, Lawrence C.","year":"1982","ISBN":"https:\/\/id.crossref.org\/isbn\/0387906223"}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/www.ams.org\/mcom\/2014-83-287\/S0025-5718-2013-02749-4\/S0025-5718-2013-02749-4.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/mcom\/2014-83-287\/S0025-5718-2013-02749-4\/S0025-5718-2013-02749-4.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,7,30]],"date-time":"2021-07-30T05:32:20Z","timestamp":1627623140000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/2014-83-287\/S0025-5718-2013-02749-4\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2013,8,20]]},"references-count":52,"journal-issue":{"issue":"287","published-print":{"date-parts":[[2014,5]]}},"alternative-id":["S0025-5718-2013-02749-4"],"URL":"https:\/\/doi.org\/10.1090\/s0025-5718-2013-02749-4","archive":["CLOCKSS","Portico"],"relation":{},"ISSN":["0025-5718","1088-6842"],"issn-type":[{"value":"0025-5718","type":"print"},{"value":"1088-6842","type":"electronic"}],"subject":[],"published":{"date-parts":[[2013,8,20]]}}}