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Prime Numbers

  • Olivier Bordellès
Part of the Universitext book series (UTX)

Abstract

This chapter is devoted to the study of the distribution of prime numbers, from Euclid to Erdős. The text follows the lines of these mathematicians who paved the way for all branches of modern number theory. After recalling the basic tools essentially due to Euclid, we investigate Chebyshev’s reasoning in his attempt to give a proof of the Prime Number Theorem. The latter will finally be shown with the theory of functions building on Riemann’s idea to extend an old-fashioned function to the whole complex plane, nowadays called the Riemann zeta-function. In the section Further Developments, we provide a proof of the PNT as a consequence of deep estimates of ζ near the line σ=1 and summation formulae. It is also the opportunity to provide explicit estimates of the classic results, yet quite rare in the literature, and to explore some of the consequences of the famous Riemann hypothesis.

Keywords

Prime Number Dirichlet Series Arithmetic Progression Critical Line Primitive Root 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. [AGP94]
    Alford WR, Granville A, Pomerance C (1994) There are infinitely many Carmichael numbers. Ann Math 140:703–722 MathSciNetCrossRefGoogle Scholar
  2. [BC09]
    Bordellès O, Cloitre B (2009) A matrix inequality for Möbius functions. JIPAM J Inequal Pure Appl Math 10, Art. 62 MathSciNetGoogle Scholar
  3. [BCY11]
    Bui HM, Conrey JB, Young MP (2011) More than 41% of the zeros of the zeta function are on the critical line. Acta Arith 150:35–64 MathSciNetzbMATHCrossRefGoogle Scholar
  4. [Bor10]
    Bordellès O (2010) The composition of the gcd and certain arithmetic functions. J Integer Seq 13, Art. 10.7.1 Google Scholar
  5. [Bru15]
    Brun V (1915) Über das Goldbachsche Gesetz und die Anzahl der Primzahlpaare. Arch Math Naturvidensk 34(8) Google Scholar
  6. [Bru19]
    Brun V (1919) La série 1/5+1/7+1/11+1/13+1/17+1/19+1/29+1/31+1/41+1/43+1/59+1/61+⋯ où les dénominateurs sont nombres premiers jumeaux est convergente ou finie. Bull Sci Math 43:124–128 Google Scholar
  7. [Bur63]
    Burgess DA (1963) On character sums and L-series, II. Proc Lond Math Soc 13:524–536 MathSciNetzbMATHCrossRefGoogle Scholar
  8. [Bur86]
    Burgess DA (1986) The character sum estimate with r=3. J Lond Math Soc (2) 33:219–226 MathSciNetzbMATHCrossRefGoogle Scholar
  9. [Bur89]
    Burgess DA (1989) Partial Gaussian sums II. Bull Lond Math Soc 21:153–158 MathSciNetzbMATHCrossRefGoogle Scholar
  10. [Bur92]
    Burgess DA (1992) Partial Gaussian sums III. Glasg Math J 34:253–261 MathSciNetzbMATHCrossRefGoogle Scholar
  11. [BV09]
    Bordellès O, Verdier N (2009) Variations autour du Postulat de Bertrand. Bull AMQ 49:25–51 Google Scholar
  12. [DE80]
    Diamond H, Erdős P (1980) On sharp elementary prime number estimates. Enseign Math 26:313–321 MathSciNetzbMATHGoogle Scholar
  13. [Dic05]
    Dickson LE (2005) History of the theory of numbers. Volume 1: Divisibility and primality. Dover, New York Google Scholar
  14. [Dus98]
    Dusart P (1998) Autour de la fonction qui compte le nombre de nombres premiers. PhD thesis, Univ. Limoges Google Scholar
  15. [Dus01]
    Dusart P (2001) Estimates for θ(x;k,l) for large values of x. Math Comput 239:1137–1168 MathSciNetGoogle Scholar
  16. [Edw74]
    Edwards HM (1974) Riemann’s zeta function. Academic Press, San Diego. 2nd edition: Dover, Mineola, New York, 2001 zbMATHGoogle Scholar
  17. [For02a]
    Ford K (2002) Vinogradov’s integral and bounds for the Riemann zeta function. Proc Lond Math Soc 85:565–633 zbMATHCrossRefGoogle Scholar
  18. [For02b]
    Ford K (2002) Zero-free regions for the Riemann zeta function. In: Bennett MA, et al. (eds) Number theory for the Millennium. AK Peters, Boston, pp 25–56 Google Scholar
  19. [GM84]
    Gupta R, Murty MR (1984) A remark on Artin’s conjecture. Invent Math 78:127–130 MathSciNetzbMATHCrossRefGoogle Scholar
  20. [Gol83]
    Goldston DA (1983) On a result of Littlewood concerning prime numbers II. Acta Arith 43:49–51 MathSciNetzbMATHGoogle Scholar
  21. [Gou04]
    Gourdon X (2004) The 1013 first zeros of the Riemann zeta function, and zeros computation at very large height. Available at: http://numbers.computation.free.fr/Constants/Miscellaneous/zetazeros1e13-1e24.pdf
  22. [GS05]
    Gupta A, Sury B (2005) Decimal expansion of 1/p and subgroup sums. Integers 5, Art. #A19 MathSciNetGoogle Scholar
  23. [Han72]
    Hanson D (1972) On the product of primes. Can Math Bull 15:33–37 MathSciNetzbMATHCrossRefGoogle Scholar
  24. [Har14]
    Hardy GH (1914) Sur les zéros de la fonction ζ(s) de Riemann. C R Acad Sci Paris 158:1012–1014 zbMATHGoogle Scholar
  25. [HB86]
    Heath-Brown DR (1986) Artin’s conjecture for primitive roots. Q J Math Oxford 37:27–38 MathSciNetzbMATHCrossRefGoogle Scholar
  26. [Hoo67]
    Hooley C (1967) On Artin’s conjecture. J Reine Angew Math 226:207–220 MathSciNetGoogle Scholar
  27. [Hux05]
    Huxley MN (2005) Exponential sums and the Riemann zeta function V. Proc Lond Math Soc 90:1–41 MathSciNetzbMATHCrossRefGoogle Scholar
  28. [Ivi85]
    Ivić A (1985) The Riemann zeta-function. Theory and applications. Wiley, New York. 2nd edition: Dover, 2003 Google Scholar
  29. [Iwa77]
    Iwaniec H (1977) The sieve of Eratosthenes–Legendre. Ann Sc Norm Super Pisa, Cl Sci 2:257–268 MathSciNetGoogle Scholar
  30. [Kad02]
    Kadiri H (2002) Une région explicite sans zéro pour les fonctions L de Dirichlet. PhD thesis, Univ. Lille 1 Google Scholar
  31. [Kad05]
    Kadiri H (2005) An explicit region without zeros for the Riemann ζ function. Acta Arith 117:303–339 MathSciNetzbMATHCrossRefGoogle Scholar
  32. [KtR06]
    Kotnik T, te Riele HJJ (2006) The Mertens conjecture revisited. In: Algorithmic number theory, 7th international symposium, ANTS-VII, Berlin, Germany, July 23–28, 2006. Lecture notes in computer science, vol 4076. Springer, Berlin, pp 156–167 CrossRefGoogle Scholar
  33. [McC84]
    McCurley KS (1984) Explicit estimates for the error term in the prime number theorem for arithmetic progressions. Math Comput 423:265–285 MathSciNetGoogle Scholar
  34. [Mon93]
    Monsky P (1993) Simplifying the proof of Dirichlet’s theorem. Am Math Mon 100:861–862 MathSciNetzbMATHCrossRefGoogle Scholar
  35. [MV07]
    Montgomery HL, Vaughan RC (2007) Multiplicative number theory Vol. I. Classical theory. Cambridge studies in advanced mathematics, vol 97 Google Scholar
  36. [Nai82]
    Nair M (1982) A new method in elementary prime number theory. J Lond Math Soc 25:385–391 zbMATHCrossRefGoogle Scholar
  37. [OtR85]
    Odlyzko A, te Riele HJJ (1985) Disproof of the Mertens conjecture. J Reine Angew Math 357:138–160 MathSciNetzbMATHGoogle Scholar
  38. [Pin87]
    Pintz J (1987) An effective disproof of the Mertens conjecture. Astérisque 147/148:325–333 MathSciNetGoogle Scholar
  39. [PS98]
    Pólya G, Szegő G (1998) Problems and theorems in analysis II. Springer, Berlin zbMATHGoogle Scholar
  40. [Ram11]
    Ramaré O (2010/2011) La méthode de Balasubramanian pour une région sans zéro. In: Groupe de Travail d’Analyse Harmonique et de Théorie Analytique des Nombres de Lille Google Scholar
  41. [Red77]
    Redheffer RM (1977) Eine explizit lösbare Optimierungsaufgabe. Int Ser Numer Math 36:203–216 MathSciNetGoogle Scholar
  42. [Rn62]
    Rosser JB, Schœnfeld L (1962) Approximate formulas for some functions of prime numbers. Ill J Math 6:64–94 zbMATHGoogle Scholar
  43. [Rn75]
    Rosser JB, Schœnfeld L (1975) Sharper bounds for the Chebyshev functions θ(x) and ψ(x). Math Comput 29:243–269 zbMATHGoogle Scholar
  44. [RR96]
    Ramaré O, Rumely R (1996) Primes in arithmetic progressions. Math Comput 65:397–425 zbMATHCrossRefGoogle Scholar
  45. [RS01]
    Rivat J, Sargos P (2001) Nombres premiers de la forme [n c]. Can J Math 53:190–209 MathSciNetCrossRefGoogle Scholar
  46. [RV08a]
    Rudelson M, Vershynin R (2008) The least singular value of a random square matrice is O(n −1/2). C R Math Acad Sci Paris 346:893–896 MathSciNetzbMATHCrossRefGoogle Scholar
  47. [RV08b]
    Rudelson M, Vershynin R (2008) The Littlewood–Offord problem and invertibility of random matrices. Adv Math 218:600–633 MathSciNetzbMATHCrossRefGoogle Scholar
  48. [Sie64]
    Sierpiński W (1964) Elementary theory of numbers. Hafner, New York. 2nd edition: North-Holland, Amsterdam, 1987 zbMATHGoogle Scholar
  49. [Sou09]
    Soundararajan K (2009) Partial sums of the Möbius function. J Reine Angew Math 631:141–152 MathSciNetzbMATHGoogle Scholar
  50. [Ste03]
    Stevenhagen P (2003) The correction factor in Artin’s primitive root conjecture. J Théor Nr Bordx 15:383–391 MathSciNetzbMATHCrossRefGoogle Scholar
  51. [Ten95]
    Tenenbaum G (1995) Introduction à la Théorie Analytique et Probabiliste des Nombres. SMF zbMATHGoogle Scholar
  52. [Tit39]
    Titchmarsh EC (1939) The theory of functions. Oxford University Press, London. 2nd edition: 1979 zbMATHCrossRefGoogle Scholar
  53. [Tit51]
    Titchmarsh EC (1951) The theory of the Riemann zeta-function. Oxford University Press, London. 2nd edition: 1986, revised by D.R. Heath-Brown zbMATHGoogle Scholar
  54. [Vau93]
    Vaughan RC (1993) On the eigenvalues of Redheffer’s matrix I. In: Number theory with an emphasis on the Markoff spectrum, Provo, Utah, 1991. Lecture notes in pure and applied mathematics, vol 147. Dekker, New York Google Scholar
  55. [Vor04]
    Voronoï G (1904) Sur une fonction transcendante et ses applications à la sommation de quelques séries. Ann Sci Éc Norm Super 21:207–268, 459–533 zbMATHGoogle Scholar
  56. [Was82]
    Washington LC (1982) Introduction to cyclotomic fields. GTM, vol 83. Springer, New York. 2nd edition: 1997 zbMATHCrossRefGoogle Scholar
  57. [Wei48a]
    Weil A (1948) On some exponential sums. Proc Natl Acad Sci USA 34:204–207 MathSciNetzbMATHCrossRefGoogle Scholar
  58. [Wei48b]
    Weil A (1948) Sur les courbes algébriques et les variétés qui s’en déduisent. Publ Inst Math Univ Strasbourg 7:1–85 MathSciNetGoogle Scholar

Copyright information

© Springer-Verlag London 2012

Authors and Affiliations

  • Olivier Bordellès
    • 1
  1. 1.AiguilheFrance

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