The distribution of square-free numbers of the form [n c ]
Journal de théorie des nombres de Bordeaux, Tome 10 (1998) no. 2, pp. 287-299.

Nous montrons que pour 1<c<61 36=1.6944⋯, la suite n c n=1,2,⋯ contient une infinité d’entiers sans facteur carré ; cela améliore un résultat antérieur dû à Rieger qui obtenait l’infinitude de ces entiers pour 1<c<1.5.

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.

Mots clés : square-free number, exponential sum, exponent pair 
@article{JTNB_1998__10_2_287_0,
     author = {Xiaodong Cao and Wenguang Zhai},
     title = {The distribution of square-free numbers of the form $[n^c]$},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {287--299},
     publisher = {Universit\'e Bordeaux I},
     volume = {10},
     number = {2},
     year = {1998},
     zbl = {0926.11066},
     mrnumber = {1828246},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/item/JTNB_1998__10_2_287_0/}
}
TY  - JOUR
AU  - Xiaodong Cao
AU  - Wenguang Zhai
TI  - The distribution of square-free numbers of the form $[n^c]$
JO  - Journal de théorie des nombres de Bordeaux
PY  - 1998
SP  - 287
EP  - 299
VL  - 10
IS  - 2
PB  - Université Bordeaux I
UR  - https://jtnb.centre-mersenne.org/item/JTNB_1998__10_2_287_0/
LA  - en
ID  - JTNB_1998__10_2_287_0
ER  - 
%0 Journal Article
%A Xiaodong Cao
%A Wenguang Zhai
%T The distribution of square-free numbers of the form $[n^c]$
%J Journal de théorie des nombres de Bordeaux
%D 1998
%P 287-299
%V 10
%N 2
%I Université Bordeaux I
%U https://jtnb.centre-mersenne.org/item/JTNB_1998__10_2_287_0/
%G en
%F JTNB_1998__10_2_287_0
Xiaodong Cao; Wenguang Zhai. The distribution of square-free numbers of the form $[n^c]$. Journal de théorie des nombres de Bordeaux, Tome 10 (1998) no. 2, pp. 287-299. https://jtnb.centre-mersenne.org/item/JTNB_1998__10_2_287_0/

[1] R.C. Baker, The square-free divisor problem. Quart. J. Math. Oxford Ser.(2) 45 (1994), no. 179, 269-277. | MR | Zbl

[2] R.C. Baker, G. Harman, J. Rivat, Primes of the form [nc]. J. Number Theory 50 (1995), 261-277. | MR | Zbl

[3] E. Bombieri, H. Iwaniec, On the order of ζ(1/2 + it). Ann. Scula Norm. Sup. Pisa Cl. Sci.(4) 13 (1986), no. 3, 449-472. | Numdam | Zbl

[4] J.-M. Deshouillers, Sur la répartition des nombers [nc] dans les progressions arithmétiques. C. R. Acad. Sci. Paris Sér. A 277 (1973), 647-650. | MR | Zbl

[5] E. Fouvry, H. Iwaniec, Exponential sums with monomials. J. Number Theory 33 (1989), 311-333. | MR | Zbl

[6] D.R. Heath-Brown, The Pjateckiĭ-Šapiro prime number theorem. J. Number Theory 16 (1983), 242-266. | MR | Zbl

[7] M.N. Huxley, M. Nair, Power free values of polynomials III. Proc. London Math. Soc.(3) 41 (1980), 66-82. | MR | Zbl

[8] H.-Q. Liu, On the number of abelian groups of given order. Acta Arith. 64 (1993), 285-296. | MR | Zbl

[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. | MR | Zbl

[10] M. Nair, Power free values of polynomials. Mathematika 23 (1976), 159-183. | MR | Zbl

[11] G.J. Rieger, Remark on a paper of Stux concerning squarefree numbers in non-linear sequences. Pacific J. Math. 78 (1978), 241-242. | MR | Zbl

[12] F. Roesler, Über die Verteilung der Primzahlen in Folgen der Form [f(n +x)]. Acta Arith. 35 (1979), 117-174. | MR | Zbl

[13] F. Roesler, Squarefree integers in non-linear sequences. Pacific J. math. 123 (1986), 223-225. | MR | Zbl

[14] B.R. Srinivasan, Lattice problems of many-dimensional hyperboloids II. Acta Arith. 37 (1963), 173-204. | MR | Zbl

[15] I.E. Stux, Distribution of squarefree integers in non-linear sequences. Pacific J. Math. 59 (1975), 577-584. | MR | Zbl

[16] J.D. Vaaler, Some extremal problems in Fourier analysis. Bull. Amer. Math. Soc. 12 (1985), 183-216. | Zbl