Algebra & Number Theory

Chebyshev's bias for products of $k$ primes

Xianchang Meng

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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

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

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

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


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.

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  • 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.