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The distribution function of a polynomial in additive functions

  • Noah Lebowitz-LockardEmail author
Article
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Abstract

A real-valued arithmetic function f is said to cluster around a point r if the upper density of inputs n for which f(n) is within \(\delta \) of r does not tend to zero as \(\delta \) goes to zero. If f does not cluster around any real number, then we say that f is nonclustering. We show that the product of nonclustering additive functions is nonclustering and provide a generalization for polynomials of nonclustering additive functions. We then use these results to prove that products of additive functions possessing continuous distribution functions also possess continuous distribution functions.

Keywords

Distribution functions Additive functions Erdős–Wintner Theorem 

Mathematics Subject Classification

11N60 

Notes

References

  1. 1.
    Elliott, P.D.T.A.: Probabilistic Number Theory I: Mean-Value Theorems, Grundlehren der mathematischen Wissenschaften, vol. 239. Springer, New York (1979)Google Scholar
  2. 2.
    Erdős, P.: On the density of some sequences of numbers. J. Lond. Math. Soc. 10, 120–125 (1935)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Erdős, P.: On the density of some sequences of numbers, II. J. Lond. Math. Soc. 12, 7–11 (1937)CrossRefGoogle Scholar
  4. 4.
    Erdős, P.: On the density of some sequences of numbers, III. J. Lond. Math. Soc. 13, 119–127 (1938)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Erdős, P., Wintner, A.: Additive arithmetical functions and statistical independence. Am. J. Math. 61, 713–721 (1939)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Galambos, J., Kátai, I.: The continuity of the limiting distribution of a function of two additive functions. Math. Z. 204, 247–252 (1990)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Halász, G.: On the distribution of additive arithmetic functions. Acta Arith. 27, 143–152 (1975)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Hall, R.R., Tenenbaum, G.: Divisors, Cambridge Tracts in Mathematics, vol. 90. Cambridge University Press, Cambridge (1988)Google Scholar
  9. 9.
    Lebowitz-Lockard, N., Pollack, P.: Clustering of linear combinations of multiplicative functions. J. Number Theory 180, 660–672 (2017)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Shapiro, H.N.: Addition of functions in probabilistic number theory. Commun. Pure Appl. Math. 26, 55–84 (1973)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Tenenbaum, G.: Introduction to Analytic and Probabilistic Number Theory, Cambridge Studies in Advanced Mathematics, vol. 46. Cambridge University Press, Cambridge (1995)Google Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.University of GeorgiaAthensUSA

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