The distribution function of a polynomial in additive functions
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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.
KeywordsDistribution functions Additive functions Erdős–Wintner Theorem
Mathematics Subject Classification11N60
- 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
- 8.Hall, R.R., Tenenbaum, G.: Divisors, Cambridge Tracts in Mathematics, vol. 90. Cambridge University Press, Cambridge (1988)Google Scholar
- 11.Tenenbaum, G.: Introduction to Analytic and Probabilistic Number Theory, Cambridge Studies in Advanced Mathematics, vol. 46. Cambridge University Press, Cambridge (1995)Google Scholar