On the Density of Coprime Tuples of the Form (n, ⌊ f1(n)⌋, …, ⌊ fk(n)⌋), Where f1, …, fk Are Functions from a Hardy Field

  • Vitaly Bergelson
  • Florian Karl Richter


Let \(k \in \mathbb{N}\) and let f1, , f k belong to a Hardy field. We prove that under some natural conditions on the k-tuple ( f1, , 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.



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. 1.Department of MathematicsThe Ohio State UniversityColumbusUSA

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