Friable values of Piatetski-Shapiro sequences
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- by Yıldırım Akbal
- Proc. Amer. Math. Soc. 145 (2017), 4255-4268
- DOI: https://doi.org/10.1090/proc/13621
- Published electronically: May 26, 2017
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Abstract:
We give various estimates for friable values of Piatetski-Shapiro sequences.References
- Yıldırım Akbal and Ahmet M. Güloğlu, Waring’s problem with Piatetski-Shapiro numbers, Mathematika 62 (2016), no. 2, 524–550. MR 3521340, DOI 10.1112/S0025579315000340
- R. C. Baker, The square-free divisor problem, Quart. J. Math. Oxford Ser. (2) 45 (1994), no. 179, 269–277. MR 1295577, DOI 10.1093/qmath/45.3.269
- Roger C. Baker, William D. Banks, Jörg Brüdern, Igor E. Shparlinski, and Andreas J. Weingartner, Piatetski-Shapiro sequences, Acta Arith. 157 (2013), no. 1, 37–68. MR 3005098, DOI 10.4064/aa157-1-3
- Cécile Dartyge, Greg Martin, and Gérald Tenenbaum, Polynomial values free of large prime factors, Period. Math. Hungar. 43 (2001), no. 1-2, 111–119. MR 1830570, DOI 10.1023/A:1015237700066
- Étienne Fouvry and Henryk Iwaniec, Exponential sums with monomials, J. Number Theory 33 (1989), no. 3, 311–333. MR 1027058, DOI 10.1016/0022-314X(89)90067-X
- S. W. Graham and G. Kolesnik, van der Corput’s method of exponential sums, London Mathematical Society Lecture Note Series, vol. 126, Cambridge University Press, Cambridge, 1991. MR 1145488, DOI 10.1017/CBO9780511661976
- Andrew Granville, Smooth numbers: computational number theory and beyond, Algorithmic number theory: lattices, number fields, curves and cryptography, Math. Sci. Res. Inst. Publ., vol. 44, Cambridge Univ. Press, Cambridge, 2008, pp. 267–323. MR 2467549
- D. R. Heath-Brown, A New $k$-th Derivative Estimate for Exponential Sums via Vinogradov’s Mean Value , arXiv:1601.04493
- D. R. Heath-Brown, The Pjateckiĭ-Šapiro prime number theorem, J. Number Theory 16 (1983), no. 2, 242–266. MR 698168, DOI 10.1016/0022-314X(83)90044-6
- Adolf Hildebrand, On the number of positive integers $\leq x$ and free of prime factors $>y$, J. Number Theory 22 (1986), no. 3, 289–307. MR 831874, DOI 10.1016/0022-314X(86)90013-2
- Adolf Hildebrand and Gérald Tenenbaum, Integers without large prime factors, J. Théor. Nombres Bordeaux 5 (1993), no. 2, 411–484. MR 1265913
- Greg Martin, An asymptotic formula for the number of smooth values of a polynomial, J. Number Theory 93 (2002), no. 2, 108–182. MR 1899301, DOI 10.1006/jnth.2001.2722
- I. I. Pyateckiĭ-Šapiro, On the distribution of prime numbers in sequences of the form $[f(n)]$, Mat. Sbornik N.S. 33(75) (1953), 559–566 (Russian). MR 0059302
- Joël Rivat and Patrick Sargos, Nombres premiers de la forme $\lfloor n^c\rfloor$, Canad. J. Math. 53 (2001), no. 2, 414–433 (French, with English summary). MR 1820915, DOI 10.4153/CJM-2001-017-0
- O. Robert and P. Sargos, Three-dimensional exponential sums with monomials, J. Reine Angew. Math. 591 (2006), 1–20. MR 2212877, DOI 10.1515/CRELLE.2006.012
- Gérald Tenenbaum, Introduction to analytic and probabilistic number theory, Cambridge Studies in Advanced Mathematics, vol. 46, Cambridge University Press, Cambridge, 1995. Translated from the second French edition (1995) by C. B. Thomas. MR 1342300
- R. C. Vaughan, A new iterative method in Waring’s problem, Acta Math. 162 (1989), no. 1-2, 1–71. MR 981199, DOI 10.1007/BF02392834
- Jie Wu, Double exponential sums and some applications, Monatsh. Math. 128 (1999), no. 3, 255–262. MR 1719419, DOI 10.1007/s006050050062
Bibliographic Information
- Yıldırım Akbal
- Affiliation: Department of Mathematics, Bilkent University, 06800 Bilkent, Ankara, Turkey
- MR Author ID: 1099434
- Email: yildirim.akbal@bilkent.edu.tr, yildirim.akbal@gmail.com
- Received by editor(s): April 11, 2016
- Received by editor(s) in revised form: November 24, 2016
- Published electronically: May 26, 2017
- Additional Notes: This work was supported by the Scientific and Technological Research Council of Turkey (114F404). We thank the referee for carefully reading the manuscript
- Communicated by: Kathrin Bringmann
- © Copyright 2017 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 145 (2017), 4255-4268
- MSC (2010): Primary 11N32; Secondary 11L03, 11B83
- DOI: https://doi.org/10.1090/proc/13621
- MathSciNet review: 3690611