The distribution of square-free numbers of the form [ n c ]

Xiaodong Cao; Wenguang Zhai

Journal de théorie des nombres de Bordeaux (1998)

  • Volume: 10, Issue: 2, page 287-299
  • ISSN: 1246-7405

Abstract

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It is proved that the sequence [ n c ] ( n = 1 , 2 , ⋯ ) contains infinite squarefree integers whenever 1 < c < 61 36 = 1 . 6944 ⋯ , which improves Rieger’s earlier range 1 < c < 1 . 5 .

How to cite

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Cao, Xiaodong, and Zhai, Wenguang. "The distribution of square-free numbers of the form $[n^c]$." Journal de théorie des nombres de Bordeaux 10.2 (1998): 287-299. <http://eudml.org/doc/248174>.

@article{Cao1998,
abstract = {It is proved that the sequence $[n^c] (n = 1, 2, \cdots )$ contains infinite squarefree integers whenever $1 &lt; c &lt; \frac\{61\}\{36\} = 1.6944 \cdots $, which improves Rieger’s earlier range $1 &lt; c &lt; 1.5$.},
author = {Cao, Xiaodong, Zhai, Wenguang},
journal = {Journal de théorie des nombres de Bordeaux},
keywords = {square-free number; exponential sum; exponent pair; distribution of square-free integers; exponential sums},
language = {eng},
number = {2},
pages = {287-299},
publisher = {Université Bordeaux I},
title = {The distribution of square-free numbers of the form $[n^c]$},
url = {http://eudml.org/doc/248174},
volume = {10},
year = {1998},
}

TY - JOUR
AU - Cao, Xiaodong
AU - Zhai, Wenguang
TI - The distribution of square-free numbers of the form $[n^c]$
JO - Journal de théorie des nombres de Bordeaux
PY - 1998
PB - Université Bordeaux I
VL - 10
IS - 2
SP - 287
EP - 299
AB - It is proved that the sequence $[n^c] (n = 1, 2, \cdots )$ contains infinite squarefree integers whenever $1 &lt; c &lt; \frac{61}{36} = 1.6944 \cdots $, which improves Rieger’s earlier range $1 &lt; c &lt; 1.5$.
LA - eng
KW - square-free number; exponential sum; exponent pair; distribution of square-free integers; exponential sums
UR - http://eudml.org/doc/248174
ER -

References

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  9. [9] H.L. Montgomery, R.C. Vaughan, The distribution of square-free numbers. in Recent Progress in Number Theory vol 1, pp. 247-256, Academic Press, London-New York, 1981. Zbl0462.10029MR637350
  10. [10] M. Nair, Power free values of polynomials. Mathematika23 (1976), 159-183. Zbl0349.10039MR429801
  11. [11] G.J. Rieger, Remark on a paper of Stux concerning squarefree numbers in non-linear sequences. Pacific J. Math.78 (1978), 241-242. Zbl0395.10049MR513296
  12. [12] F. Roesler, Über die Verteilung der Primzahlen in Folgen der Form [f(n +x)]. Acta Arith.35 (1979), 117-174. Zbl0342.10027MR547671
  13. [13] F. Roesler, Squarefree integers in non-linear sequences. Pacific J. math.123 (1986), 223-225. Zbl0562.10017MR834150
  14. [14] B.R. Srinivasan, Lattice problems of many-dimensional hyperboloids II. Acta Arith.37 (1963), 173-204. Zbl0118.28201MR153641
  15. [15] I.E. Stux, Distribution of squarefree integers in non-linear sequences. Pacific J. Math.59 (1975), 577-584. Zbl0297.10033MR387218
  16. [16] J.D. Vaaler, Some extremal problems in Fourier analysis. Bull. Amer. Math. Soc.12 (1985), 183-216. Zbl0575.42003

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