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Multiplicative Arithmetic Functions of Several Variables: A Survey

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Mathematics Without Boundaries

Abstract

We survey general properties of multiplicative arithmetic functions of several variables and related convolutions, including the Dirichlet convolution and the unitary convolution. We introduce and investigate a new convolution, called gcd convolution. We define and study the convolutes of arithmetic functions of several variables, according to the different types of convolutions. We discuss the multiple Dirichlet series and Bell series and present certain arithmetic and asymptotic results of some special multiplicative functions arising from problems in number theory, group theory, and combinatorics. We give a new proof to obtain the asymptotic density of the set of ordered r-tuples of positive integers with pairwise relatively prime components and consider a similar question related to unitary divisors.

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References

  1. Alkan, E., Zaharescu, A., Zaki, M.: Arithmetical functions in several variables. Int. J. Number Theory 1, 383–399 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  2. Alkan, E., Zaharescu, A., Zaki, M.: Unitary convolution for arithmetical functions in several variables. Hiroshima Math. J. \bf36 113–124 (2006)

    MATH  MathSciNet  Google Scholar 

  3. Alkan, E., Zaharescu, A., Zaki, M.: Multidimensional averages and Dirichlet convolution. Manuscr. Math. 123, 251–267 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  4. Apostol, T.M.: Introduction to Analytic Number Theory. Springer, New York (1976)

    Google Scholar 

  5. Buschman, R.G.: lcm-products of number-theoretic functions revisited. Kyungpook Math. J. 39, 159–164 (1999)

    MATH  MathSciNet  Google Scholar 

  6. Cai, J.-Y. Bach, E.: On testing for zero polynomials by a set of points with bounded precision. Theor. Comp. Sci. 296, 15–25 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  7. Carroll, T.B., Gioia, A.A.: On a subgroup of the group of multiplicative arithmetic functions. J. Aust. Math. Soc. 20(Series A), 348–358 (1975)

    Article  MATH  MathSciNet  Google Scholar 

  8. Casas, O.: Arithmetical functions in two variables. An analogue of a result of Delange. Lect. Mat. 27, 5–12 (2006). (Número especial)

    MathSciNet  Google Scholar 

  9. Cashwell, E.D., Everett, C.J.: The ring of number-theoretic functions. Pac. J. Math. 9, 975–985 (1959)

    Article  MATH  MathSciNet  Google Scholar 

  10. Cesàro, E.: Étude moyenne du plus grand commun diviseur de deux nombres. Annali di Matematica Pura ed Applicata 13, 235–250 (1885)

    Article  MATH  Google Scholar 

  11. Chan, T.H., Kumchev, A.V.: On sums of Ramanujan sums. Acta Arith. 152, 1–10 (2012)

    Google Scholar 

  12. Christopher, J.: The asymptotic density of some k dimensional sets. Amer. Math. Mon. 63, 399–401 (1956)

    Article  MATH  MathSciNet  Google Scholar 

  13. Cohen, E.: Arithmetical functions of a greatest common divisor, III. Cesàro’s divisor problem. Proc. Glasg. Math. Assoc. 5, 67–75 (1961–1962)

    Article  Google Scholar 

  14. Dehaye, P.-O.: On the structure of the group of multiplicative arithmetical functions. Bull. Belg. Math. Soc. 9, 15–21 (2002)

    MATH  MathSciNet  Google Scholar 

  15. Delange, H.: On some sets of pairs of positive integers. J. Number Theory 1, 261–279 (1969)

    Article  MATH  MathSciNet  Google Scholar 

  16. Delange, H.: Sur les fonctions multiplicatives de plusieurs entiers. Enseignement Math. (2)16 (1970). (219–246 (1971), Errata: (2) 17 (1971), 186)

    Google Scholar 

  17. Delany, J.E.: Groups of arithmetical functions. Math. Mag. 78, 83–97 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  18. Diaconis, P., Erdős, P.: On the Distribution of the greatest common divisor, in A festschrift for Herman Rubin, IMS lecture notes monograph series institute of mathematical statistics, vol 45, (2004), 56–61 (original version: Technical Report No. 12, Department of Statistics, Stanford University, Stanford, 1977)

    Google Scholar 

  19. Fernández, J.L., Fernández, P.: Asymptotic normality and greatest common divisors, Preprint, arXiv:1302.2357 (2013) [math.PR]

    Google Scholar 

  20. Fernández, J.L., Fernández, P.: On the probability distribution of the gcd and lcm of r-tuples of integers, Preprint, arXiv:1305.0536 (2013) [math.NT]

    Google Scholar 

  21. Fernández, J.L., Fernández, P.: Equidistribution and coprimality, Preprint, arXiv:1310.3802 (2013) [math.NT]

    Google Scholar 

  22. Fernández, J.L., Fernández, P.: Random index of codivisibility, Preprint, arXiv:1310.4681v1 (2013) [math.NT]

    Google Scholar 

  23. Finch, S.: Modular forms on \(\rm SL_2(\mathbb Z)\), manuscript. http://www.people.fas.harvard.edu/ sfinch/csolve/frs.pdf (2005)

    Google Scholar 

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

    Article  MATH  Google Scholar 

  25. Hampejs, M., Tóth, L.: On the subgroups of finite Abelian groups of rank three. Ann. Univ. Sci. Bp. Sect. Comp. 39, 111–124 (2013)

    MATH  Google Scholar 

  26. Hampejs, M., Holighaus, N., Tóth, L., Wiesmeyr, C.: On the subgroups of the group \(\mathbb Z_m \times\mathbb Z_n\), Preprint, arXiv:1211.1797 (2012) [math.GR]

    Google Scholar 

  27. Hardy, G.H., Wright, E.M.: An Introduction to the Theory of Numbers, 6th edn. In: Heath-Brown D. R., Silverman, J. H. (eds.) Oxford University Press, Oxford (2008)

    Google Scholar 

  28. Haukkanen, P.: Derivation of arithmetical functions under the Dirichlet convolution, manuscript.

    Google Scholar 

  29. Heppner, E.: Über benachbarte multiplikative zahlentheoretische Funktionen mehrerer Variablen. Arch. Math. (Basel) 35, 454–460 (1980)

    Article  MATH  MathSciNet  Google Scholar 

  30. Heppner, E.: Über Mittelwerte multiplikativer zahlentheoretischer Funktionen mehrerer Variablen. Monatshefte Math. 91, 1–9 (1981)

    Article  MATH  MathSciNet  Google Scholar 

  31. Hu, J.: The probability that random positive integers are k-wise relatively prime. Int. J. Number Theory 9, 1263–1271 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  32. Huxley, M.N.: Exponential sums and lattice points III. Proc. Lond. Math. Soc. 87, 591–609 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  33. Indlekofer K.-H.: Multiplikative Funktionen mehrerer Variablen. J. Reine Angew. Math. 256, 180–184 (1972)

    MATH  MathSciNet  Google Scholar 

  34. Johnson, K.R.: Reciprocity in Ramanujan’s sum. Math. Mag. 59, 216–222 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  35. Krätzel, E., Nowak, W.G., Tóth, L.: On certain arithmetic functions involving the greatest common divisor. Cent. Eur. J. Math. 10, 761–774 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  36. Kühleitner, M., Nowak, W.G.: On a question of A. Schinzel: Omega estimates for a special type of arithmetic functions. Cent. Eur. J. Math. 11, 477–486 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  37. Lehmer, D.H.: A new calculus of numerical functions. Amer. J. Math. 53(4), 843–854 (1931)

    Article  MathSciNet  Google Scholar 

  38. Lehmer, D.H.: On a theorem of von Sterneck. Bull. Am. Math. Soc. 37(10), 723–726 (1931)

    Article  MathSciNet  Google Scholar 

  39. Lehmer, D.H.: Arithmetic of double series. Trans. Am. Math. Soc. 33, 945–957 (1931)

    Article  MathSciNet  Google Scholar 

  40. Liskovets, V.A.: A multivariate arithmetic function of combinatorial and topological significance. Integers 10, 155–177 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  41. MacHenry, T.: A subgroup of the group of units in the ring of arithmetic functions. Rocky Mt. J. Math. 29, 1055–1065 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  42. McCarthy, P.J.: Introduction to Arithmetical Functions. Springer, New York (1986)

    Google Scholar 

  43. Narkiewicz, W.: On a summation formula of E. Cohen. Colloq. Math. 11, 85–86 (1963)

    MATH  MathSciNet  Google Scholar 

  44. Nowak, W.G., Tóth, L.: On the average number of subgroups of the group \(\mathbb Z_m \times\mathbb Z_n\), Int. J. Number Theory, 10, 363–374 (2014)(Accepted, arXiv:1307.1414 [math.NT])

    Google Scholar 

  45. Onozuka, T.: The multiple Dirichlet product and the multiple Dirichlet series, manuscript, 2013

    Google Scholar 

  46. Pakapongpun, A., Ward, T.: Functorial orbit counting. J. Integer Seq. 12, 2–0 (2009) (Article 09.2. 4)

    MathSciNet  Google Scholar 

  47. Peter, M.: Rekurrente zahlentheoretische Funktionen in mehreren Variablen. Arch. Math. (Basel) 68, 202–213 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  48. Ryden, R.W.: Groups of arithmetic functions under Dirichlet convolution. Pac. J. Math. 44, 355–360 (1973)

    Article  MATH  MathSciNet  Google Scholar 

  49. Sándor, J., Bege, A.: The Möbius function: Generalizations and extensions. Adv. Stud. Contemp. Math. (Kyungshang), 6(2), 77–128 (2003)

    MATH  MathSciNet  Google Scholar 

  50. Sándor, J., Crstici, B.: Handbook of Number Theory II. Kluwer Academic Publishers, Dordrecht (2004)

    Google Scholar 

  51. Scheid, H.: Einige Ringe zahlentheoretischer Funktionen. J. Reine Angew. Math. 237, 1–11 (1969)

    MATH  MathSciNet  Google Scholar 

  52. Schwab, E.D.: Generalized arithmetical functions of three variables. Int. J. Number Theory 6, 1689–1699 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  53. Schwarz, W., Spilker, J.: Arithmetical functions, An introduction to elementary and analytic properties of arithmetic functions and to some of their almost-periodic properties, London Mathematical Society Lecture Note Series, 184. Cambridge University Press, Cambridge (1994)

    Google Scholar 

  54. Shapiro, H.N.: On the convolution ring of arithmetic functions. Comm. Pure Appl. Math. 25, 287–336 (1972)

    Article  MATH  MathSciNet  Google Scholar 

  55. Sivaramakrishnan, R.: Classical Theory of Arithmetic Functions, Monographs and Textbooks in Pure and Applied Mathematics, vol. 126. Marcel Dekker (1989)

    Google Scholar 

  56. Sloane, N.J.A.: The On-Line Encyclopedia of Integer Sequences. http://oeis.org

  57. Smith, D.A.: Generalized arithmetic function algebras. In: Gioia, A.A., Goldsmith, D.L. (eds.) The Theory of Arithmetic Functions, vol. 251, pp. 205–245. Springer, Berlin (1972) (Proceedings of the Conference at Western Michigan University, Kalamazoo, Mich., 1971, Lecture Notes in Math)

    Google Scholar 

  58. Snellman, J.: Truncations of the ring of arithmetical functions with unitary convolution. Int. J. Math. Game Theory Algebra 13, 485–519 (2003)

    MATH  MathSciNet  Google Scholar 

  59. Snellman, J.: The ring of arithmetical functions with unitary convolution: Divisorial and topological properties. Arch. Math. (Brno), 40, 161–179 (2004)

    MATH  MathSciNet  Google Scholar 

  60. Subbarao, M.V.: On some arithmetic convolutions, In: The Theory of Arithmetic Functions, Lecture Notes in Mathematics No. 251, 247–271. Springer (1972)

    Google Scholar 

  61. Tóth, L.: On the asymptotic densities of certain subsets of \(\mathbb N^k\). Riv. Mat. Univ. Parma (6)4 121–131 (2001)

    Google Scholar 

  62. Tóth, L.: The probability that k positive integers are pairwise relatively prime. Fibonacci Quart. 40, 13–18 (2002)

    MATH  MathSciNet  Google Scholar 

  63. Tóth, L.: A survey of gcd-sum functions. J. Integer Seq. 13, 2–3 (2010). (Article 10.8. 1)

    Google Scholar 

  64. Tóth, L.: Menon’s identity and arithmetical sums representing functions of several variables. Rend. Sem. Mat. Univ. Politec. Torino 69, 97–110 (2011)

    MATH  Google Scholar 

  65. Tóth, L.: On the number of cyclic subgroups of a finite Abelian group. Bull. Math. Soc. Sci. Math. Roum. (N.S.) 55(103), 423–428 (2012)

    Google Scholar 

  66. Tóth, L.: Some remarks on a paper of V. A. Liskovets. Integers 12, 97–111 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  67. Tóth, L.: Sums of products of Ramanujan sums. Ann. Univ. Ferrara 58, 183–197 (2012)

    Article  Google Scholar 

  68. Tóth, L.: Two generalizations of the Busche-Ramanujan identities. Int. J. Number Theory 9, 1301–1311 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  69. Tóth, L.: A survey of the alternating sum-of-divisors function. Acta Univ. Sapientiae Math. 5, 93–107 (2013) (accepted, arXiv:1111.4842 [math.NT])

    Google Scholar 

  70. Tóth, L.: Averages of Ramanujan sums: Note on two papers by E. Alkan, accepted, arXiv:1305.6018 [math.NT].

    Google Scholar 

  71. Tóth, L., Haukkanen, P.: On the binomial convolution of arithmetical functions. J. Comb. Number Theory 1, 31–47 (2009)

    MATH  MathSciNet  Google Scholar 

  72. Ushiroya, N.: Mean-value theorems for multiplicative arithmetic functions of several variables. Integers 12, 989–1002 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  73. Vaidyanathaswamy, R.: The theory of multiplicative arithmetic functions. Trans. Am. Math. Soc. 33, 579–662 (1931)

    Article  MathSciNet  Google Scholar 

  74. von Sterneck, R.D.: Ableitung zahlentheoretischer Relationen mit Hilfe eines mehrdimensionalen Systemes von Gitterpunkten. Monatsh. Math. Phys. 5, 255–266 (1894)

    Article  MATH  MathSciNet  Google Scholar 

  75. Zaharescu, A., Zaki, M.: Derivations and generating degrees in the ring of arithmetical functions. Proc. Indian Acad. Sci. (Math. Sci.) 117(2), 167–175 (2007)

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Institute of Mathematics, Universität für Bodenkultur, Gregor Mendel-Straße 33, A-1180, Vienna, Austria

    László Tóth

  2. Department of Mathematics, University of Pécs, Ifjúság u. 6, H-7624, Pécs, Hungary

    László Tóth

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  1. László Tóth
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  1. Department of Mathematics, National Technical University of Athens, Athens, Greece

    Themistocles M. Rassias

  2. Dept. Industrial & Systems Engineering, University of Florida, Gainesville, Florida, USA

    Panos M. Pardalos

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Tóth, L. (2014). Multiplicative Arithmetic Functions of Several Variables: A Survey. In: Rassias, T., Pardalos, P. (eds) Mathematics Without Boundaries. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-1106-6_19

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