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Arithmetic Functions

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Arithmetic Tales

Part of the book series: Universitext ((UTX))

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Abstract

This chapter may be viewed as an extension of the previous one, in the sense that multiplicative functions may generalize Euclid’s fundamental theorem of integer factorizations. The text is aimed at introducing the Dirichlet convolution product, thus giving a ring structure to the set of arithmetic functions, and then establishing some useful summation results for multiplicative functions with the help of the Möbius inversion formula. The section Further Developments is devoted to a complete study of Dirichlet series from an arithmetic viewpoint and we also provide some estimates for other types of summation, such as multiplicative functions over short intervals or additive functions. Finally, a brief account of Selberg’s sieve and the large sieve is also given.

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Notes

  1. 1.

    Some authors also use the notation δ or i.

  2. 2.

    Some authors also use the notation d k .

  3. 3.

    There is no official notation for this function in the literature. For instance, Ivić [Ivi85] uses the notation f k .

  4. 4.

    It is noteworthy that \(\tau_{k} ( p^{\alpha}) = \mathcal{D}_{(1,\dotsc,1)} (\alpha)\) where the vector \((1,\dotsc,1)\) has k components. See also Proposition 7.118.

  5. 5.

    See [Tit39, Theorem 2.8].

  6. 6.

    This result is due to Pincherle.

  7. 7.

    A slowly varying function is a non-zero Lebesgue measurable function L:[x 0,+∞[⟶ℂ for some x 0>0 for which

    $$\lim_{x \rightarrow\infty} \frac{L(cx)}{L(x)} = 1 $$

    for any c>0.

  8. 8.

    The best inequalities to date for θ are \(\frac{1}{4} \leqslant \theta\leqslant\frac{131}{416}\). See Chap. 6.

References

  1. Bombieri E (1974) Le Grand Crible dans la Théorie Analytique des Nombres, vol 18. Société Mathématique de France, Paris

    Google Scholar 

  2. Bordellès O (2009) Le problème des diviseurs de Dirichlet. Quadrature 71:21–30

    Article  MATH  Google Scholar 

  3. Bordellès O (2010) The composition of the gcd and certain arithmetic functions. J Integer Seq 13, Art. 10.7.1

    Google Scholar 

  4. Erdős P (1957) Some unsolved problems. Mich Math J 4:291–300

    Article  Google Scholar 

  5. Friedlander J, Iwaniec H (2005) Summation formulæ for coefficients of L-functions. Can J Math 57:494–505

    Article  MathSciNet  MATH  Google Scholar 

  6. Friedlander J, Iwaniec H (2010) Opera de Cribro. Colloquium publications, vol 57. Am. Math. Soc., Providence

    MATH  Google Scholar 

  7. Gallagher PX (1967) The large sieve. Mathematika 14:14–20

    Article  MathSciNet  MATH  Google Scholar 

  8. Gould HW (1972) Combinatorial identities. A standardized set of tables listing 500 binomial coefficients summations. Morgantown, West Virginia

    Google Scholar 

  9. Gupta H (1978) Finite differences of the partition function. Math Comput 32:1241–1243

    Article  MATH  Google Scholar 

  10. Halász G (1968) Über die Mittelwerte multiplikativer zahlentheoretischer Funktionen. Acta Math Acad Sci Hung 19:365–403

    Article  MATH  Google Scholar 

  11. Halberstam H, Richert H-E (1979) On a result of R.R. Hall. J Number Theory 11:76–89

    Article  MathSciNet  MATH  Google Scholar 

  12. Halberstam H, Richert H-E (2011) Sieve methods. Dover, Mineola

    Google Scholar 

  13. Hall RR, Tenenbaum G (1988) Divisors. Cambridge University Press, Cambridge

    Book  MATH  Google Scholar 

  14. Ivić A (1985) The Riemann zeta-function. Theory and applications. Wiley, New York. 2nd edition: Dover, 2003

    Google Scholar 

  15. Iwaniec H, Kowalski E (2004) Analytic number theory. Colloquium publications, vol 53. Am. Math. Soc., Providence

    MATH  Google Scholar 

  16. Krätzel E (1970) Die maximale Ordnung der Anzahl der wesentlich verschiedenen Abelschen Gruppen n-ter Ordnung. Q J Math (2) Oxford Ser 21:273–275

    Article  MATH  Google Scholar 

  17. Kubilius J (1971) The method of Dirichlet generating series in the theory of distribution of additive arithmetic functions. I. Liet Mat Rink 11:125–134

    MathSciNet  MATH  Google Scholar 

  18. Landau E (1927) Einführung in die elementare und analytische Theorie der algebraischen Zahlen und der Ideale. Teubner, Leipzig. 2nd edition: Chelsea, 1949

    MATH  Google Scholar 

  19. Levin BV, Fainleb AS (1967) Application of some integral equations to problems of number theory. Russ Math Surv 22:119–204

    Article  MATH  Google Scholar 

  20. Linnik JV (1941) The large sieve. Dokl Akad Nauk SSSR 30:292–294

    MathSciNet  Google Scholar 

  21. Marraki ME (1995) Fonction sommatoire de la fonction de Möbius, 3. Majorations effectives forte. J Théor Nr Bordx 7:407–433

    Article  MATH  Google Scholar 

  22. Martin G (2002) An asymptotic formula for the number of smooth values of a polynomial. J Number Theory 93:108–182

    Article  MathSciNet  MATH  Google Scholar 

  23. Montgomery HL (1978) The analytic principle of the large sieve. Bull Am Math Soc 84:547–567

    Article  MATH  Google Scholar 

  24. Motohashi Y (1979) A note on Siegel’s zeros. Proc Jpn Acad 55:190–192

    Article  MathSciNet  MATH  Google Scholar 

  25. Montgomery HL, Vaughan RC (1973) The large sieve. Mathematika 20:119–134

    Article  MathSciNet  MATH  Google Scholar 

  26. Nair M, Tenenbaum G (1998) Short sums of certain arithmetic functions. Acta Math 180:119–144

    Article  MathSciNet  MATH  Google Scholar 

  27. Parson A, Tull J (1978) Asymptotic behavior of multiplicative functions. J Number Theory 10:395–420

    Article  MathSciNet  MATH  Google Scholar 

  28. Shiu P (1980) A Brun–Titchmarsh theorem for multiplicative functions. J Reine Angew Math 313:161–170

    MathSciNet  MATH  Google Scholar 

  29. Sivaramakrishnan R (1989) Classical theory of arithmetical functions. Pure and applied mathematics, vol 126. Dekker, New York

    Google Scholar 

  30. Skrabutenas R (1974) Asymptotic expansion of sums of multiplicative functions. Liet Mat Rink 14:115–126

    MathSciNet  MATH  Google Scholar 

  31. Schwarz L, Spilker J (1994) Arithmetical functions. Cambridge University Press, Cambridge

    MATH  Google Scholar 

  32. Suryanarayana D (1971) The number of k-free divisors of an integer. Acta Arith 47:345–354

    MathSciNet  Google Scholar 

  33. Sargos P, Wu J (2000) Multiple exponential sums with monomials and their applications in number theory. Acta Math Hung 88:333–354

    Article  MathSciNet  Google Scholar 

  34. Titchmarsh EC (1939) The theory of functions. Oxford University Press, London. 2nd edition: 1979

    Book  MATH  Google Scholar 

  35. Tuljaganova MI (1978) A generalization of a theorem of Halász. Izv Akad Nauk SSSR 4:35–40

    MathSciNet  Google Scholar 

  36. Vaughan RC (1973) Some applications of Montgomery’s sieve. J Number Theory 5:641–679

    Article  MathSciNet  Google Scholar 

  37. Voronoï G (1904) Sur une fonction transcendante et ses applications à la sommation de quelques séries. Ann Sci Éc Norm Super 21:207–268

    MATH  Google Scholar 

  38. Wirsing E (1961) Das asymptotische Verhalten vun Summen über multiplikative Funktionen. Math Ann 143:75–102

    Article  MathSciNet  MATH  Google Scholar 

  39. Wu J (1995) Problèmes de diviseurs exponentiels et entiers exponentiellement sans facteur carré. J Théor Nr Bordx 7:133–141

    Article  MATH  Google Scholar 

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  1. allée de la Combe 2, Aiguilhe, France

    Olivier Bordellès

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  1. Olivier Bordellès
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Bordellès, O. (2012). Arithmetic Functions. In: Arithmetic Tales. Universitext. Springer, London. https://doi.org/10.1007/978-1-4471-4096-2_4

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