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

Chapter

Abstract

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.

Notes

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.

References

  1. 1.
    S. Abramovich, Y.Y. Nikitin, On the probability of co-primality of two natural numbers chosen at random (Who was the first to pose and solve this problem?). arXiv e-prints (2016). http://arxiv.org/abs/1608.05435
  2. 2.
    V. Bergelson, G. Kolesnik, Y. Son, Uniform distribution of subpolynomial functions along primes and applications. J. Anal. Math. arXiv e-prints (2015, to appear). http://arxiv.org/abs/1503.04960
  3. 3.
    M. Boshernitzan, An extension of Hardy’s class L of “orders of infinity”. J. Anal. Math. 39, 235–255 (1981)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    M. Boshernitzan, New “orders of infinity”. J. Anal. Math. 41, 130–167 (1982)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    M.D. Boshernitzan, Uniform distribution and Hardy fields. J. Anal. Math. 62, 225–240 (1994)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    E. Cesàro, Questions proposées, # 75. Mathesis 1, 184 (1981)Google Scholar
  7. 7.
    E. Cesàro, Solutions de questions proposées, question 75. Mathesis 3, 224–225 (1983)Google Scholar
  8. 8.
    T. Cochrane, Trigonometric approximation and uniform distribution modulo one. Proc. Am. Math. Soc. 103, 695–702 (1988)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    F. Delmer, J.-M. Deshouillers, On the probability that n and [n c] are coprime. Period. Math. Hungar. 45, 15–20 (2002)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    G. Dirichlet, Mathematische Werke. Band II, Herausgegeben auf Veranlassung der Königlich Preussischen Akademie der Wissenschaften von L. Kronecker, Druck und Verlag von Georg Reimer (1897)Google Scholar
  11. 11.
    M. Drmota, R.F. Tichy, Sequences, Discrepancies and Applications. Lecture Notes in Mathematics, vol. 1651 (Springer, Berlin, 1997)Google Scholar
  12. 12.
    P. Erdős, G.G. Lorentz, On the probability that n and g(n) are relatively prime. Acta Arith. 5, 35–44 (1959)MathSciNetMATHGoogle Scholar
  13. 13.
    T. Estermann, On the number of primitive lattice points in a parallelogram. Can. J. Math. 5, 456–459 (1953)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    N. Frantzikinakis, Equidistribution of sparse sequences on nilmanifolds. J. Anal. Math. 109, 353–395 (2009)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    P.J. Grabner, Erdős-Turán type discrepancy bounds. Monatsh. Math. 111, 127–135 (1991)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    S.W. Graham, G. Kolesnik, Van der Corput’s Method of Exponential Sums. London Mathematical Society Lecture Note Series, vol. 126 (Cambridge University Press, Cambridge, 1991)Google Scholar
  17. 17.
    G.H. Hardy, Properties of logarithmico-exponential functions. Proc. Lond. Math. Soc. S2–10, 54–90 (1912)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    G.H. Hardy, Orders of Infinity. The Infinitärcalcül of Paul du Bois-Reymond (Hafner Publishing, New York, 1971). Reprint of the 1910 edition, Cambridge Tracts in Mathematics and Mathematical Physics, No. 12Google Scholar
  19. 19.
    J.F. Koksma, Some theorems on Diophantine inequalities, Scriptum no. 5, Math. Centrum Amsterdam (1950)Google Scholar
  20. 20.
    J. Lambek, L. Moser, On integers n relatively prime to f(n). Can. J. Math. 7, 155–158 (1955)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    F. Mertens, Über einige asymptotische Gesetze der Zahlentheorie. J. Reine Angew. Math. 77, 289–338 (1874)MathSciNetMATHGoogle Scholar
  22. 22.
    J. Spilker, Die Fastperiodizität der Watson-Funktion. Arch. Math. (Basel) 74, 26–29 (2000)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    J.J. Sylvester, The Collected Mathematical Papers. Volume III (1870–1883) (Cambridge University Press, Cambridge, 1909), pp. 672–676MATHGoogle Scholar
  24. 24.
    J.J. Sylvester, The Collected Mathematical Papers. Volume IV (1882–1897) (Cambridge University Press, Cambridge, 1912), pp. 84–87Google Scholar
  25. 25.
    P. Szüsz, Über ein Problem der Gleichverteilung, in Comptes Rendus du Premier Congrès des Mathématiciens Hongrois, 27 Août–2 Septembre 1950 (Akadémiai Kiadó, Budapest, 1952), pp. 461–472Google Scholar
  26. 26.
    J.G. van der Corput, Neue zahlentheoretische Abschätzungen. Math. Ann. 89, 215–254 (1923)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    J.G. van der Corput, Neue zahlentheoretische Abschätzungen (Zweite Mitteilung). Math. Z. 29, 397–426 (1929)MathSciNetCrossRefGoogle Scholar
  28. 28.
    G.L. Watson, On integers n relatively prime to [αn]. Can. J. Math. 5, 451–455 (1953)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Department of MathematicsThe Ohio State UniversityColumbusUSA

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