Frontiers of Mathematics in China >

2017 , Vol. 12 >Issue 6: 1515 - 1525

DOI: https://doi.org/10.1007/s11464-017-0652-1

RESEARCH ARTICLE

Distribution of cube-free numbers with form [nc]

  • Min ZHANG ,
  • Jinjiang LI
Expand
  • Department of Mathematics, China University of Mining and Technology, Beijing 100083, China

Received date: 22 Feb 2017

Accepted date: 05 Jun 2017

Published date: 27 Nov 2017

Copyright

2017 Higher Education Press and Springer-Verlag GmbH Germany
Fold

Abstract

We prove that there are infinite cube-free numbers of the form [nc] for any fixed real number c ∈ (1, 11/6).

Key words： Cube-free number; exponential sum; asymptotic formula

Cite this article

Min ZHANG , Jinjiang LI . Distribution of cube-free numbers with form [nc][J]. Frontiers of Mathematics in China, 2017 , 12(6) : 1515 -1525 . DOI: 10.1007/s11464-017-0652-1

References

Publishing order | Descend order by publishing year | Descend order by cited within
1
BakerR C, BamksWD, BrüdernJ, ShparlinskiI E, WeingartnerA J. Piatetski-Shapiro sequences. Acta Arith, 2013, 157(1): 37–68

DOI

2
BalogA. On a variant of the Piatetski-Shapiro prime number theorem. Groupe de Travail en Théorie Analytique et Elementaire des Nombres 1987–1988. Publications Mathématiques Orsay 89-01. Orsay: Univ de Paris XI, 1989, 3–11

3
BurievK. Additive Problems with Prime Numbers. Thesis. Moscow: Moscow Univ, 1989 (in Russian)

4
CaoX D, ZhaiW G. The distribution of square-free numbers of the form [nc]. J Théor Nombres Bordeaux, 1998, 10(2): 287–299

DOI

5
CaoX D, ZhaiW G. Multiple exponential sums with monomials. Acta Arith, 2000, 92(3): 195–213

6
CaoX D, ZhaiW G. On the distribution of square-free numbers of the form [nc](II). Acta Math Sinica (Chin Ser), 2008, 51(6): 1187–1194 (in Chinese)

7
DeshouillersJ M. Sur la repartition des mombers [nc]dans les progressions arithmetiques. C R Acad Sci Paris Śer A, 1973, 277: 647–650

8
GrahamS W, KolesnikG. Van der Corput’s Method of Exponential Sums. London Math Soc Lecture Note Ser, 126. Cambridge: Cambridge Univ Press, 1991

DOI

9
Heath-BrownD R. The Pjatecki˘ı-˘Sapiro prime number theorem. J Number Theory, 1983, 16(2): 242–266

DOI

10
Piatetski-ShapiroI I. On the distribution of prime numbers in sequences of the form [f(m)]. Mat Sb (N S), 1953, 33(75)(3): 559–566 (in Russian)

11
RiegerG J. Remark on a paper of Stux concerning squarefree numbers in non-linear sequences. Pacific J Math, 1978, 78(1): 241–242

DOI

12
RivatJ, WuJ. Prime numbers of the form [nc]. Glasg Math J, 2001, 43: 237–254

DOI

13
RobertO, SargosP. Three-dimensional exponential sums with monomials. J Reine Angew Math, 2006, 591: 1–20

DOI

14
StuxI E. Distribution of squarefree integers in non-linear sequences. Pacific J Math, 1975, 59(2): 577–584

DOI

15
VaalerJ D. Some extremal functions in Fourier analysis. Bull Amer Math Soc, 1985, 12(2): 183–216

Options
Outlines

/

〈 〉