Algebra & Number Theory

Chebyshev's bias for products of $k$ primes

Xianchang Meng

Full-text: Access denied (no subscription detected)

However, an active subscription may be available with MSP at msp.org/ant.

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

For any k1, we study the distribution of the difference between the number of integers nx with ω(n)=k or Ω(n)=k in two different arithmetic progressions, where ω(n) is the number of distinct prime factors of n and Ω(n) is the number of prime factors of n counted with multiplicity. Under some reasonable assumptions, we show that, if k is odd, the integers with Ω(n)=k have preference for quadratic nonresidue classes; and if k is even, such integers have preference for quadratic residue classes. This result confirms a conjecture of Richard Hudson. However, the integers with ω(n)=k always have preference for quadratic residue classes. Moreover, as k increases, the biases become smaller and smaller for both of the two cases.

Article information

Source
Algebra Number Theory, Volume 12, Number 2 (2018), 305-341.

Dates
Received: 6 October 2016
Revised: 26 September 2017
Accepted: 30 October 2017
First available in Project Euclid: 23 May 2018

Permanent link to this document
https://projecteuclid.org/euclid.ant/1527040846

Digital Object Identifier
doi:10.2140/ant.2018.12.305

Mathematical Reviews number (MathSciNet)
MR3803705

Zentralblatt MATH identifier
06880890

Subjects
Primary: 11M26: Nonreal zeros of $\zeta (s)$ and $L(s, \chi)$; Riemann and other hypotheses
Secondary: 11M06: $\zeta (s)$ and $L(s, \chi)$ 11N60: Distribution functions associated with additive and positive multiplicative functions

Keywords
Chebyshev's bias Dirichlet $L$-function Hankel contour generalized Riemann hypothesis

Citation

Meng, Xianchang. Chebyshev's bias for products of $k$ primes. Algebra Number Theory 12 (2018), no. 2, 305--341. doi:10.2140/ant.2018.12.305. https://projecteuclid.org/euclid.ant/1527040846


Export citation

References

  • P. L. Chebyshev, “Lettre de M. le professeur Tchébyshev á M. Fuss, sur un nouveau théoreme rélatif aux nombres premiers contenus dans la formes $4n+1$ et $4n+3$”, Bull. Classe Phys. Acad. Imp. Sci. St. Petersburg 11 (1853), 208.
  • H. Davenport, Multiplicative number theory, 3rd ed., Graduate Texts in Mathematics 74, Springer, 2000.
  • P. G. L. Dirichlet, “Beweis des Satzes, dass jede unbegrenzte arithmetische Progression, deren erstes Glied und Differenz ganze Zahlen ohne gemeinschaftlichen Factor sind, unendlich viele Primzahlen enthält”, Abh. K önig. Preuss. Akad. 34 (1837), 45–81. Reprinted as pp. 313–342 in his Werke, vol. 1, Reimer, Berlin, 1889 and Chelsea, New York, 1969.
  • D. Fiorilli and G. Martin, “Inequities in the Shanks–Rényi prime number race: an asymptotic formula for the densities”, J. Reine Angew. Math. 676 (2013), 121–212.
  • K. Ford and S. Konyagin, “Chebyshev's conjecture and the prime number race”, pp. 67–91 in IV International Conference “Modern Problems of Number Theory and its Applications”: Current Problems, Part II (Russian) (Tula, 2001), edited by V. N. Chubarikov and G. I. Arkhipov, Mosk. Gos. Univ. im. Lomonosova, Mekh.-Mat. Fak., Moscow, 2002.
  • K. Ford and J. Sneed, “Chebyshev's bias for products of two primes”, Experiment. Math. 19:4 (2010), 385–398.
  • A. Granville and G. Martin, “Prime number races”, Amer. Math. Monthly 113:1 (2006), 1–33.
  • R. H. Hudson, “A common combinatorial principle underlies Riemann's formula, the Chebyshev phenomenon, and other subtle effects in comparative prime number theory, I”, J. Reine Angew. Math. 313 (1980), 133–150.
  • A. A. Karatsuba, Basic analytic number theory, Springer, 1993.
  • S. Knapowski and P. Turán, “Comparative prime-number theory, I. Introduction”, Acta Math. Acad. Sci. Hungar. 13 (1962), 299–314.
  • Y. Lamzouri, “Prime number races with three or more competitors”, Math. Ann. 356:3 (2013), 1117–1162.
  • E. Landau, Handbuch der Lehre von der Verteilung der Primzahlen, 2 Bände, Leipzig B.G. Teubner, Berlin, 1909. Reprinted as 3rd edition, Chelsea, 1974.
  • Y. K. Lau and J. Wu, “Sums of some multiplicative functions over a special set of integers”, Acta Arith. 101:4 (2002), 365–394.
  • J. Leech, “Note on the distribution of prime numbers”, J. London Math. Soc. 32 (1957), 56–58.
  • J. E. Littlewood, “Sur la distribution des nombres premiers”, C. R. Acad. Sci., Paris 158 (1914), 1869–1872.
  • I. G. Macdonald, Symmetric functions and Hall polynomials, 2nd ed., The Clarendon Press, Oxford University Press, New York, 1995.
  • A. Mendes and J. Remmel, Counting with symmetric functions, Developments in Mathematics 43, Springer, 2015.
  • M. Rubinstein and P. Sarnak, “Chebyshev's bias”, Experiment. Math. 3:3 (1994), 173–197.
  • G. Tenenbaum, Introduction à la théorie analytique et probabiliste des nombres, 2nd ed., Cours Spécialisés [Specialized Courses] 1, Société Mathématique de France, Paris, 1995.
  • E. C. Titchmarsh, The theory of the Riemann zeta-function, 2nd ed., The Clarendon Press, Oxford University Press, New York, 1986.