Values of the Euler $\phi$-function not divisible by a given odd prime, and the distribution of Euler-Kronecker constants for cyclotomic fields
HTML articles powered by AMS MathViewer
- by Kevin Ford, Florian Luca and Pieter Moree PDF
- Math. Comp. 83 (2014), 1447-1476 Request permission
Abstract:
Let $\phi$ denote Euler’s phi function. For a fixed odd prime $q$ we investigate the first and second order terms of the asymptotic series expansion for the number of $n\leqslant x$ such that $q\nmid \phi (n)$. 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 $k$-tuples and a conjecture of Ihara about the distribution of these Euler-Kronecker constants cannot be both true.References
- George E. Andrews and Bruce C. Berndt, Ramanujan’s lost notebook. Part III, Springer, New York, 2012. MR 2952081, DOI 10.1007/978-1-4614-3810-6
- A. I. Badzyan, The Euler-Kronecker constant, Mat. Zametki 87 (2010), no. 1, 35–47 (Russian, with Russian summary); English transl., Math. Notes 87 (2010), no. 1-2, 31–42. MR 2730381, DOI 10.1134/S0001434610010050
- Bruce C. Berndt and Ken Ono, Ramanujan’s unpublished manuscript on the partition and tau functions with proofs and commentary, Sém. Lothar. Combin. 42 (1999), Art. B42c, 63. The Andrews Festschrift (Maratea, 1998). MR 1701582
- Yann Bugeaud, Maurice Mignotte, and Samir Siksek, Classical and modular approaches to exponential Diophantine equations. I. Fibonacci and Lucas perfect powers, Ann. of Math. (2) 163 (2006), no. 3, 969–1018. MR 2215137, DOI 10.4007/annals.2006.163.969
- Harold Davenport, Multiplicative number theory, 3rd ed., Graduate Texts in Mathematics, vol. 74, Springer-Verlag, New York, 2000. Revised and with a preface by Hugh L. Montgomery. MR 1790423
- Hubert Delange, Sur des formules de Atle Selberg, Acta Arith. 19 (1971), 105–146. (errata insert) (French). MR 289432, DOI 10.4064/aa-19-2-105-146
- Christopher Deninger, On the analogue of the formula of Chowla and Selberg for real quadratic fields, J. Reine Angew. Math. 351 (1984), 171–191. MR 749681, DOI 10.1515/crll.1984.351.171
- L.E. Dickson, A new extension of Dirichlet’s theorem on prime numbers, Messenger of Math., 33 (1904), 155–161.
- P. D. T. A. Elliott and H. Halberstam, A conjecture in prime number theory, Symposia Mathematica, Vol. IV (INDAM, Rome, 1968/69) Academic Press, London, 1970, pp. 59–72. MR 0276195
- P. Erdős, On a problem of G. Golomb, J. Austral. Math. Soc. 2 (1961/1962), 1–8. MR 0123539, DOI 10.1017/S144678870002632X
- FFTW Fast Fourier Transform C Library, available at http://www.fftw.org/.
- Greg Fee and Andrew Granville, The prime factors of Wendt’s binomial circulant determinant, Math. Comp. 57 (1991), no. 196, 839–848. MR 1094948, DOI 10.1090/S0025-5718-1991-1094948-8
- Tony Forbes, Prime clusters and Cunningham chains, Math. Comp. 68 (1999), no. 228, 1739–1747. MR 1651752, DOI 10.1090/S0025-5718-99-01117-5
- T. Forbes, Prime $k$-tuples, http://anthony.d.forbes.googlepages.com/ktuplets.htm
- A. Granville, On the size of the first factor of the class number of a cyclotomic field, Invent. Math. 100 (1990), no. 2, 321–338. MR 1047137, DOI 10.1007/BF01231189
- Kevin Ford, The distribution of integers with a divisor in a given interval, Ann. of Math. (2) 168 (2008), no. 2, 367–433. MR 2434882, DOI 10.4007/annals.2008.168.367
- A. Fröhlich and M. J. Taylor, Algebraic number theory, Cambridge Studies in Advanced Mathematics, vol. 27, Cambridge University Press, Cambridge, 1993. MR 1215934
- H. Halberstam and H.-E. Richert, Sieve methods, London Mathematical Society Monographs, No. 4, Academic Press [Harcourt Brace Jovanovich, Publishers], London-New York, 1974. MR 0424730
- G.H. Hardy and J.E. Littlewood, Some problems of a “Partitio Numerorum”: III On the expression of a number as a sum of primes, Acta Math. 44 (1922), 1–70.
- Yasufumi Hashimoto, Yasuyuki Iijima, Nobushige Kurokawa, and Masato Wakayama, Euler’s constants for the Selberg and the Dedekind zeta functions, Bull. Belg. Math. Soc. Simon Stevin 11 (2004), no. 4, 493–516. MR 2115723
- Douglas Hensley and Ian Richards, On the incompatibility of two conjectures concerning primes, Analytic number theory (Proc. Sympos. Pure Math., Vol. XXIV, St. Louis Univ., St. Louis, Mo., 1972) Amer. Math. Soc., Providence, R.I., 1973, pp. 123–127. MR 0340194
- Yasutaka Ihara, On the Euler-Kronecker constants of global fields and primes with small norms, Algebraic geometry and number theory, Progr. Math., vol. 253, Birkhäuser Boston, Boston, MA, 2006, pp. 407–451. MR 2263195, DOI 10.1007/978-0-8176-4532-8_{5}
- Yasutaka Ihara, The Euler-Kronecker invariants in various families of global fields, Arithmetics, geometry, and coding theory (AGCT 2005), Sémin. Congr., vol. 21, Soc. Math. France, Paris, 2010, pp. 79–102 (English, with English and French summaries). MR 2856562
- Yasutaka Ihara, V. Kumar Murty, and Mahoro Shimura, On the logarithmic derivatives of Dirichlet $L$-functions at $s=1$, Acta Arith. 137 (2009), no. 3, 253–276. MR 2496464, DOI 10.4064/aa137-3-6
- E. Landau, Über 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), 305–312. (See also his Collected Papers.)
- Edmund Landau, Losung des Lehmer’schen Problems, Amer. J. Math. 31 (1909), no. 1, 86–102 (German). MR 1506062, DOI 10.2307/2370180
- E. Landau, Handbuch der Lehre von der Verteilung der Primzahlen, 3rd ed., Chelsea, New York, 1953.
- A. Languasco and A. Zaccagnini, A note on Mertens’ formula for arithmetic progressions, J. Number Theory 127 (2007), no. 1, 37–46. MR 2351662, DOI 10.1016/j.jnt.2006.12.015
- Alessandro Languasco and Alessandro Zaccagnini, On the constant in the Mertens product for arithmetic progressions. I. Identities. part 1, Funct. Approx. Comment. Math. 42 (2010), no. part 1, 17–27. MR 2640766, DOI 10.7169/facm/1269437065
- A. Languasco and A. Zaccagnini, On the constant in the Mertens product for arithmetic progressions. II. Numerical values, Math. Comp. 78 (2009), no. 265, 315–326. MR 2448709, DOI 10.1090/S0025-5718-08-02148-0
- Alessandro Languasco and Alessandro Zaccagnini, Computing the Mertens and Meissel-Mertens constants for sums over arithmetic progressions, Experiment. Math. 19 (2010), no. 3, 279–284. With an appendix by Karl K. Norton. MR 2743571, DOI 10.1080/10586458.2010.10390624
- Philippe Lebacque, Generalised Mertens and Brauer-Siegel theorems, Acta Arith. 130 (2007), no. 4, 333–350. MR 2365709, DOI 10.4064/aa130-4-3
- E. M. Matveev, An explicit lower bound for a homogeneous rational linear form in logarithms of algebraic numbers. II, Izv. Ross. Akad. Nauk Ser. Mat. 64 (2000), no. 6, 125–180 (Russian, with Russian summary); English transl., Izv. Math. 64 (2000), no. 6, 1217–1269. MR 1817252, DOI 10.1070/IM2000v064n06ABEH000314
- Kevin S. McCurley, Explicit estimates for the error term in the prime number theorem for arithmetic progressions, Math. Comp. 42 (1984), no. 165, 265–285. MR 726004, DOI 10.1090/S0025-5718-1984-0726004-6
- H. L. Montgomery and R. C. Vaughan, The large sieve, Mathematika 20 (1973), 119–134. MR 374060, DOI 10.1112/S0025579300004708
- Hugh L. Montgomery and Robert C. Vaughan, Multiplicative number theory. I. Classical theory, Cambridge Studies in Advanced Mathematics, vol. 97, Cambridge University Press, Cambridge, 2007. MR 2378655
- Pieter Moree, On some claims in Ramanujan’s ‘unpublished’ manuscript on the partition and tau functions, Ramanujan J. 8 (2004), no. 3, 317–330. MR 2111687, DOI 10.1007/s11139-004-0142-4
- Pieter Moree, Chebyshev’s bias for composite numbers with restricted prime divisors, Math. Comp. 73 (2004), no. 245, 425–449. MR 2034131, DOI 10.1090/S0025-5718-03-01536-9
- P. Moree, Values of the Euler phi function not divisible by a prescribed odd prime, math.NT/0611509, 2006, unpublished preprint.
- Pieter Moree, Counting numbers in multiplicative sets: Landau versus Ramanujan, Math. Newsl. 21 (2011), no. 3, 73–81. MR 3012680
- V. Kumar Murty, The Euler-Kronecker constant of a cyclotomic field, Ann. Sci. Math. Québec 35 (2011), no. 2, 239–247 (English, with English and French summaries). MR 2917834
- Władysław Narkiewicz, Elementary and analytic theory of algebraic numbers, 2nd ed., Springer-Verlag, Berlin; PWN—Polish Scientific Publishers, Warsaw, 1990. MR 1055830
- OEIS Foundation (2011), The On-Line Encyclopedia of Integer Sequences, http://oeis.org/.
- Michael Rosen, A generalization of Mertens’ theorem, J. Ramanujan Math. Soc. 14 (1999), no. 1, 1–19. MR 1700882
- J. Barkley Rosser and Lowell Schoenfeld, Approximate formulas for some functions of prime numbers, Illinois J. Math. 6 (1962), 64–94. MR 137689
- J. Barkley Rosser and Lowell Schoenfeld, Sharper bounds for the Chebyshev functions $\theta (x)$ and $\psi (x)$, Math. Comp. 29 (1975), 243–269. MR 457373, DOI 10.1090/S0025-5718-1975-0457373-7
- Daniel Shanks, The second-order term in the asymptotic expansion of $B(x)$, Math. Comp. 18 (1964), 75–86. MR 159174, DOI 10.1090/S0025-5718-1964-0159174-9
- Blair K. Spearman and Kenneth S. Williams, Values of the Euler phi function not divisible by a given odd prime, Ark. Mat. 44 (2006), no. 1, 166–181. MR 2237219, DOI 10.1007/s11512-005-0001-6
- Gérald Tenenbaum, Introduction to analytic and probabilistic number theory, Cambridge Studies in Advanced Mathematics, vol. 46, Cambridge University Press, Cambridge, 1995. Translated from the second French edition (1995) by C. B. Thomas. MR 1342300
- M. A. Tsfasman, Asymptotic behaviour of the Euler-Kronecker constant, Algebraic geometry and number theory, Progr. Math., vol. 253, Birkhäuser Boston, Boston, MA, 2006, pp. 453–458. MR 2263196, DOI 10.1007/978-0-8176-4532-8_{6}
- C.J. de la Vallée-Poussin, Recherches analytiques sur la théorie des nombres premiers I, Ann. Soc. Sci. Bruxelles 20 (1896), 183–256.
- Lawrence C. Washington, Introduction to cyclotomic fields, Graduate Texts in Mathematics, vol. 83, Springer-Verlag, New York, 1982. MR 718674, DOI 10.1007/978-1-4684-0133-2
Additional Information
- Kevin Ford
- Affiliation: Department of Mathematics, University of Illinois at Urbana-Champaign, 1409 West Green Street, Urbana, Illinois 61801, USA
- MR Author ID: 325647
- ORCID: 0000-0001-9650-725X
- Email: ford@math.uiuc.edu
- Florian Luca
- Affiliation: Fundación Marcos Moshinsky, UNAM, Circuito Exterior, C.U., Apdo. Postal 70-543, Mexico D.F. 04510, Mexico
- MR Author ID: 630217
- Email: fluca@matmor.unam.mx
- Pieter Moree
- Affiliation: Max-Planck-Institut für Mathematik, Vivatsgasse 7, D-53111 Bonn, Germany.
- MR Author ID: 290905
- Email: moree@mpim-bonn.mpg.de
- Received by editor(s): January 17, 2012
- Received by editor(s) in revised form: June 20, 2012, and August 22, 2012
- Published electronically: August 20, 2013
- Additional Notes: The first author was supported in part by National Science Foundation grants DMS-0555367 and DMS-0901339.
The second author was supported in part by Grants SEP-CONACyT 79685 and PAPIIT 100508 - © Copyright 2013 American Mathematical Society
- Journal: Math. Comp. 83 (2014), 1447-1476
- MSC (2010): Primary 11N37, 11Y60
- DOI: https://doi.org/10.1090/S0025-5718-2013-02749-4
- MathSciNet review: 3167466