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The distribution function of a polynomial in additive functions

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Abstract

A real-valued arithmetic function f is said to cluster around a pointr 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.

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  1. University of Georgia, Athens, GA, USA

    Noah Lebowitz-Lockard

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  1. Noah Lebowitz-Lockard
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Correspondence to Noah Lebowitz-Lockard.

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Lebowitz-Lockard, N. The distribution function of a polynomial in additive functions. Ramanujan J 49, 491–504 (2019). https://doi.org/10.1007/s11139-018-0117-5

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  • DOI: https://doi.org/10.1007/s11139-018-0117-5

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