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Rough values of Piatetski-Shapiro sequences

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Abstract

An integer is called y-rough if it is composed solely of primes \(> y\). Let \(\lfloor {.}\rfloor \) be the floor function. In this paper, we exhibit an asymptotic formula for the counting function of integers \(n \leqslant x\) such that \(\lfloor {n^c}\rfloor \) is y-rough uniformly for a range of y that depends on \(1< c < 2229/1949\).

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References

  1. Baker, R.C.: The square-free divisor problem. Q. J. Math. Oxf. Ser. (2) 45(179), 269–277 (1994)

  2. Baker, R.C.: Diophantine inequalities. Lond. Math. Soc. Monogr. (N.S.) 1 (Clarendon Press, New York) (1986)

  3. Fouvry, E., Iwaniec, H.: Exponential sums with monomials. J. Number Theory 33(3), 311–333 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  4. Heath-Brown, D.R.: A New \(k\)-th Derivative Estimate for Exponential Sums via Vinogradov’s Mean Value. arXiv:1601.04493

  5. Graham, S.W.,  Kolesnik, G.: van der Corput’s method of exponential sums. In: London Mathematical Society Lecture Note Series, vol. 126. Cambridge University Press, Cambridge (1991) (ISBN 0-521-33927-8)

  6. Heath-Brown, D.R.: The Pjatecki–Sapiro prime number theorem. J. Number Theory 16(2), 242–266 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  7. Hildebrand, A.: On the number of positive integers \(\le x\) and free of prime factors \({\>}y\). J. Number Theory 22, 289–307 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  8. Piatetski-Shapiro, I.I.: On the distribution of prime numbers in sequences of the form\(\left\lfloor {f(n)}\right\rfloor \) (Russian). Mat. Sb. N.S. 33 (75), 559–566 (1953)

  9. Rivat, J., Sargos, P.: Nombres premiers de la forme \(\left\lfloor {n^c}\right\rfloor \). Can. J. Math 53(2), 414–433 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  10. Rivat, J., Wu, J.: Prime numbers of the form \(\left\lfloor {n^c}\right\rfloor \). Glasg. Math. J. 43(2), 237–254 (2001)

  11. Robert, O., Sargos, P.: Three-dimensional exponential sums with monomials. J. Reine Angew. Math. 591, 1–20 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  12. Tenenbaum, G.: Introduction to analytic and probabilistic number theory. In: Cambridge Studies in Advanced Mathematics, vol. 46. Cambridge University Press, Cambridge (1995) (ISBN: 0-521-41261-7)

  13. Vaughan, R.C.: A new iterative method in Waring’s problem. Acta Math. 162(1–2), 1–71 (1989)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Bilkent University, Bilkent, 06800, Ankara, Turkey

    Yıldırım Akbal

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  1. Yıldırım Akbal
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Correspondence to Yıldırım Akbal.

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Communicated by J. Schoißengeier.

During the preparation of the paper the author was supported by TÜBİTAK Research Grant No. 114F404.

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Akbal, Y. Rough values of Piatetski-Shapiro sequences. Monatsh Math 185, 1–15 (2018). https://doi.org/10.1007/s00605-016-0993-y

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  • DOI: https://doi.org/10.1007/s00605-016-0993-y

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