Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



Chebyshev's bias for composite numbers with restricted prime divisors

Author: Pieter Moree
Journal: Math. Comp. 73 (2004), 425-449
MSC (2000): Primary 11N37, 11Y60; Secondary 11N13
Published electronically: May 21, 2003
MathSciNet review: 2034131
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $\pi(x;d,a)$ denote the number of primes $p\le x$with $p\equiv a(\operatorname{mod} d)$. Chebyshev's bias is the phenomenon for which ``more often'' $\pi(x;d,n)>\pi(x;d,r)$, than the other way around, where $n$ is a quadratic nonresidue mod $d$ and $r$ is a quadratic residue mod $d$. If $\pi(x;d,n)\ge \pi(x;d,r)$ for every $x$ up to some large number, then one expects that $N(x;d,n)\ge N(x;d,r)$ for every $x$. Here $N(x;d,a)$ denotes the number of integers $n\le x$such that every prime divisor $p$ of $n$ satisfies $p\equiv a(\operatorname{mod}d)$. In this paper we develop some tools to deal with this type of problem and apply them to show that, for example, $N(x;4,3)\ge N(x;4,1)$ for every $x$.

In the process we express the so-called second order Landau-Ramanujan constant as an infinite series and show that the same type of formula holds for a much larger class of constants.

References [Enhancements On Off] (What's this?)

  • 1. C. Bays and R.H. Hudson, Details of the first region of integers $x$ with $\pi_{3,2}(x)<\pi_{3,1}(x)$, Math. Comp. 32 (1978), 571-576. MR 57:16175
  • 2. B.C. Berndt and K. Ono, Ramanujan's unpublished manuscript on the partition and tau functions with proofs and commentary, The Andrews Festschrift (Maratea, 1998), (Eds.) D. Foata and G.N. Han, 2001, 39-110. MR 2000i:01027
  • 3. S. Bochner and B. Jessen, Distribution functions and positive definite functions, Ann. Math. 35 (1934), 252-257.
  • 4. P.L. Chebyshev, Lettre de M. le Professeur Tchébychev à M. Fuss sur un nouveaux théorème relatif aux nombres premiers contenus dans les formes $4n+1$ ets $4n+3$, Bull. Classe Phys. Acad. Imp. Sci. St. Petersburg 11 (1853), 208.
  • 5. H. Cohen, Advanced topics in computational number theory, GTM 193, Springer-Verlag, New York, 2000. MR 2000k:11144
  • 6. H. Cohen, High precision computation of Hardy-Littlewood constants, draft of a preprint, available at${}^{\sim}$cohen/.
  • 7. H. Cohen and F. Dress, Estimations numériques du reste de la fonction sommatoire relative aux entiers sans facteur carré, Publ. Math. Orsay 88/02 (1988), 73-76. MR 89i:11103
  • 8. J.H. Conway and N.J.A. Sloane, Sphere packings, lattices and groups, Third edition, Springer-Verlag, New York, 1999. MR 2000b:11077
  • 9. D.A. Cox, The arithmetic-geometric mean of Gauss, Enseign. Math. 30 (1984), 275-330. MR 86a:01027
  • 10. H. Davenport, Multiplicative number theory, Third revised edition, Springer-Verlag, New York, 2000. MR 2001f:11001
  • 11. K. Dilcher, Generalized Euler constants for arithmetical progressions, Math. Comp. 59 (1992), 259-282. MR 92k:11145
  • 12. P. Dusart, Autour de la fonction qui compte le nombre de nombres premiers, PhD thesis, Université de Limoges, 1998.
  • 13. S.R. Finch, Mathematical Constants, Cambridge University Press, (2003); website
  • 14. K. Ford and R.H. Hudson, Sign changes in $\pi_{q,a}(x)-\pi_{q,b}(x)$, Acta Arith. 100 (2001), 297-314.
  • 15. J. Kaczorowski, The boundary values of generalized Dirichlet series and a problem of Chebyshev, Astérisque 209 (1992), 227-235. MR 94a:11144
  • 16. J. Kaczorowski, Boundary values of Dirichlet series and the distribution of primes, European Congress of Mathematics, Vol. I (Budapest, 1996), 237-254, Progr. Math., 168, Birkhäuser, Basel, 1998. MR 99f:11119
  • 17. S. Knapowski and P. Turán, Further developments in the comparative prime-number theory. I, Acta Arith. 9 (1964), 23-40. MR 29:75
  • 18. E. Landau, Über die zu einem algebraischen Zahlkörper gehörige Zetafunktion und die Ausdehnung der Tschebyschefschen Primzahlentheorie aus das Problem der Verteilung der Primideale, J. Reine Angew. Math. 125 (1903), 64-188.
  • 19. E. Landau, Über die Einteilung der positiven ganzen Zahlen in vier Klassen nach der mindest Anzahl der zu ihrer additiven Zusammensetzung erforderlichen Quadrate, Arch. der Math. und Phys. 13 (1908), 305-312.
  • 20. B.V. Levin and A.S. Fainleib, Application of some integral equations to problems of number theory, Russian Math. Surveys 22 (1967), 119-204. MR 37:5174
  • 21. J.E. Littlewood, Distribution des nombres premiers, C.R. Acad. Sci. Paris 158 (1914), 1869-1872.
  • 22. S.D. Miller and G. Moore, Landau-Siegel zeroes and black hole entropy, Asian J. Math. 4 (2000), 183-211. MR 2002i:11063
  • 23. P. Moree, Approximation of singular series and automata, Manuscripta Math. 101 (2000), 385-399. MR 2001f:11204
  • 24. P. Moree, On some claims in Ramanujan's `unpublished' manuscript on the partition and tau functions, arXiv:math.NT/0201265, to appear in The Ramanujan Journal.
  • 25. P. Moree and J. Cazaran, On a claim of Ramanujan in his first letter to Hardy, Exposition. Math. 17 (1999), 289-311. MR 2001c:11103
  • 26. P. Moree and H.J.J. te Riele, The hexagonal versus the square lattice, arXiv:math.NT/0204332, to appear in Math. Comp.
  • 27. L. Moser and R.A. MacLeod, The error term for the squarefree integers, Canad. Math. Bull. 9 (1966), 303-306. MR 34:150
  • 28. W. Narkiewicz, The development of prime number theory. From Euclid to Hardy and Littlewood, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2000. MR 2001c:11098
  • 29. V. Nevanlinna, On constants connected with the prime number theorem for arithmetic progressions, Ann. Acad. Sci. Fenn. Ser. A I 539 (1973), 11 pp. MR 49:7219
  • 30. A.G. Postnikov, Introduction to analytic number theory, AMS translations of mathematical monographs 68, AMS, Providence, Rhode Island, 1988. MR 89a:11001
  • 31. O. Ramaré, Sur un théorème de Mertens, Manuscripta Math. 108 (2002), 495-513.
  • 32. O. Ramaré and R. Rumely, Primes in arithmetic progressions, Math. Comp. 65 (1996), 397-425. MR 97a:11144
  • 33. J.B. Rosser and L. Schoenfeld, Approximate formulas for some functions of prime numbers, Illinois Journal Math. 6 (1962), 64-94. MR 25:1139
  • 34. M. Rubinstein and P. Sarnak, Chebyshev's bias, Experiment. Math. 3 (1994), 173-197. MR 96d:11099
  • 35. P. Schmutz Schaller, Geometry of Riemann surfaces based on closed geodesics, Bull. Amer. Math. Soc. (N.S.) 35 (1998), 193-214. MR 99b:11098
  • 36. L. Schoenfeld, Sharper bounds for the Chebyshev functions $\theta(x)$ and $\psi(x)$. II, Math. Comp. 30 (1976), 337-360. MR 56:15881b
  • 37. J.-P. Serre, Divisibilité de certaines fonctions arithmétiques, Enseign. Math. 22 (1976), 227-260. MR 55:7958
  • 38. D. Shanks, The second order term in the asymptotic expansion of $B(x)$, Math. Comp. 18 (1964), 75-86. MR 28:2391
  • 39. J.M. Song, Sums over nonnegative multiplicative functions over integers without large prime factors. I, Acta Arith. 97 (2001), 329-351. MR 2002f:11130
  • 40. P. Turán, On a new method of analysis and its applications, John Wiley and Sons, Inc., New York, 1984. MR 86b:11059
  • 41. P. Turán, Collected papers of Paul Turán, Vol. 1-3, Ed. P. Erdös. Akadémiai Kiadó, Budapest, 1990. MR 91i:01145
  • 42. K.S. Williams, Mertens' theorem for arithmetic progressions, J. Number Theory 6 (1974), 353-359. MR 51:392
  • 43. A. Wintner, Upon a statistical method in the theory of diophantine approximations, Amer. J. Math. 55 (1933), 309-331.
  • 44. E. Wirsing, Das asymptotische Verhalten von Summen über multiplikative Funktionen, Math. Ann. 143 (1961), 75-102. MR 24:A1241

Similar Articles

Retrieve articles in Mathematics of Computation with MSC (2000): 11N37, 11Y60, 11N13

Retrieve articles in all journals with MSC (2000): 11N37, 11Y60, 11N13

Additional Information

Pieter Moree
Affiliation: KdV Institute, University of Amsterdam, Plantage Muidergracht 24, 1018 TV Amsterdam, The Netherlands

Keywords: Comparative number theory, constants, primes in progression, multiplicative functions
Received by editor(s): November 7, 2001
Received by editor(s) in revised form: May 2, 2002
Published electronically: May 21, 2003
Article copyright: © Copyright 2003 American Mathematical Society

American Mathematical Society