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On the Density of Coprime Tuples of the Form (n, ⌊ f 1(n)⌋, …, ⌊ f k (n)⌋), Where f 1, …, f k Are Functions from a Hardy Field

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Number Theory – Diophantine Problems, Uniform Distribution and Applications

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Abstract

Let \(k \in \mathbb{N}\) and let f 1, …, f k belong to a Hardy field. We prove that under some natural conditions on the k-tuple ( f 1, …, f k ) the density of the set

$$\displaystyle{\big\{n \in \mathbb{N}:\gcd (n,\lfloor \,f_{1}(n)\rfloor,\ldots,\lfloor \,f_{k}(n)\rfloor ) = 1\big\}}$$

exists and equals \(\frac{1} {\zeta (k+1)}\), where ζ is the Riemann zeta function.

It is a well-known theorem of Čebyšev1 that the probability of the relation gcd(n, m) = 1 is \(\frac{6} {\pi ^{2}}\). One can expect this still to remain true if m = g(n) is a function of n, provided that g(n) does not preserve arithmetic properties of n.

P. Erdős and G. Lorentz

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Notes

  1. 1.

    The attribution of this result to Čebyšev (Chebyshëv) seems not to be justified; see, however, the very interesting recent preprint [1] where Čebyšev’s role in the popularization of this theorem is traced and analyzed. The result itself goes back to Dirichlet (see [10, pp. 51–66] where the equivalent statement \(\sum _{n=1}^{N}\phi (n) \sim \frac{3} {\pi ^{2}} n^{2}\) is proven) and was rediscovered multiple times—see, for example, [6, 7, 21, 23, 24]. It is worth noting that it was Cesàro who formulated this result in probabilistic terms [6] and also gave a probabilistic, though not totally rigorous, proof in [7].

  2. 2.

    We define a germ at ∞ to be any equivalence class of functions under the equivalence relationship \((\,f \sim g) \Leftrightarrow \big (\exists t_{0}> 0\ \text{such that}\ f(t) = g(t)\ \text{for all}\ t \in [t_{0},\infty )\big)\).

  3. 3.

    By a logarithmico-exponential function we mean any function \(f: (0,\infty ) \rightarrow \mathbb{R}\) that can be obtained from constants, log(t) and exp(t) using the standard arithmetical operations +, −, ⋅ , ÷, and the operation of composition.

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Acknowledgements

The authors would like to thank the anonymous referees for their helpful comments and Christian Elsholtz for the efficient handling of the submission process.The first author gratefully acknowledges the support of the NSF under grant DMS-1500575.

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Authors and Affiliations

  1. Department of Mathematics, The Ohio State University, Columbus, OH, 43210, USA

    Vitaly Bergelson & Florian Karl Richter

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  1. Vitaly Bergelson
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  2. Florian Karl Richter
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Correspondence to Vitaly Bergelson .

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Editors and Affiliations

  1. Institute of Analysis and Number Theory, Graz University of Technology, Graz, Austria

    Christian Elsholtz

  2. Institute of Analysis and Number Theory, Graz University of Technology, Graz, Austria

    Peter Grabner

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Bergelson, V., Richter, F.K. (2017). On the Density of Coprime Tuples of the Form (n, ⌊ f 1(n)⌋, …, ⌊ f k (n)⌋), Where f 1, …, f k Are Functions from a Hardy Field. In: Elsholtz, C., Grabner, P. (eds) Number Theory – Diophantine Problems, Uniform Distribution and Applications. Springer, Cham. https://doi.org/10.1007/978-3-319-55357-3_5

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