Annali di Matematica Pura ed Applicata

, Volume 101, Issue 1, pp 153–169 | Cite as

Prime-independent arithmetical functions

  • J. Knopfmacher
  • J. N. Ridley
Article

Summary

This paper studies arithmetical functions that are prime-independent and multiplicative. Firstly, it establishes necessary and sufficient conditions for such functions to possess simple formulae relating them to the zeta function. Then it investigates asymptotic average-values and moments of such functions. The results apply to functions of ideals in algebraic numbers fields, or of isomorphism classes in certain categories, as well as to functions of positive integers.

Keywords

Positive Integer Zeta Function Simple Formula Isomorphism Class Number Field 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    P. T. Bateman - H. G. Diamond,Asymptotic distribution of Beurling's generalized prime numbers, Studies in Number Theory, MAA Studies in Mathematics,6 (1969).Google Scholar
  2. [2]
    P. Erdös -A. Rényi,On the mean value of nonnegative multiplicative number-theoretical functions, Michigan J. Math.,12 (1965), pp. 321–338.CrossRefMATHGoogle Scholar
  3. [3]
    P. Erdös -G. Szekeres,Über die Anzahl der Abelschen Gruppen gegebener Ordnung und über ein verwandtes zahlentheoretisches Problem, Acta Sci. Math. Szeged,7 (1935), pp. 95–102.Google Scholar
  4. [4]
    T. Estermann,On certain functions represented by Dirichlet series, Proc. London Math. Soc.,27 (1928), pp. 435–448.MATHGoogle Scholar
  5. [5]
    G. H. Hardy -S. Ramanujan,Asymptotic formulae concerning the distribution of integers of various types, Proc. London Math. Soc.,16 (1917), pp. 112–132.Google Scholar
  6. [6]
    G. H. Hardy - E. M. Wright,An Introduction to the Theory of Numbers, Oxford, 1954.Google Scholar
  7. [7]
    M. Kac,Note on the distribution of values of the arithmetic function d(m), Bulletin American Math. Soc.,47 (1941), pp. 815–817.MATHMathSciNetCrossRefGoogle Scholar
  8. [8]
    D. G. Kendall -R. A. Rankin,On the number of abelian groups of a given order, Quart. J. Math. (Oxford Series),18 (1947), pp. 197–208.MathSciNetGoogle Scholar
  9. [9]
    J. Knopfmacher,Arithmetical properties of finite rings and algebras, and analytic number theory, I–VI, J. reine angew. Math.,252 (1972), pp. 16–43;254 (1972), pp. 74–99;259 (1973), pp. 157–170, and forthcoming.MATHMathSciNetGoogle Scholar
  10. [10]
    E. Landau,Einführung in die elementare und analytische Theorie der algebraischen Zahlen und der Ideale, New York, 1949.Google Scholar
  11. [11]
    E. C. Titchmarsh,Some problems in the analytic theory of numbers, Quart. J. Math. (Oxford Series),13 (1942), pp. 129–152.MATHMathSciNetGoogle Scholar
  12. [12]
    B. M. Wilson,Proofs of some formulae enunciated by Ramanujan, Proc. London Math. Soc.,21 (1922), pp. 235–255.MATHGoogle Scholar
  13. [13]
    O. Zariski - P. Samuel,Commutative Algebra, New York, 1960.Google Scholar

Copyright information

© Fondazione Annali di Matematica Pura ed Applicata 1974

Authors and Affiliations

  • J. Knopfmacher
    • 1
  • J. N. Ridley
    • 1
  1. 1.Johannesburg

Personalised recommendations