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.

DOI : https://doi.org/10.5802/jtnb.229
Mots clés: square-free number, exponential sum, exponent pair
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     author = {Cao, Xiaodong and Zhai, Wenguang},
     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},
     doi = {10.5802/jtnb.229},
     zbl = {0926.11066},
     mrnumber = {1828246},
     language = {en},
     url = {jtnb.centre-mersenne.org/item/JTNB_1998__10_2_287_0/}
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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. doi : 10.5802/jtnb.229. 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 1295577 | Zbl 0812.11053

[2] R.C. Baker, G. Harman, J. Rivat, Primes of the form [nc]. J. Number Theory 50 (1995), 261-277. | MR 1316821 | Zbl 0822.11062

[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 0615.10047

[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 337834 | Zbl 0265.10024

[5] E. Fouvry, H. Iwaniec, Exponential sums with monomials. J. Number Theory 33 (1989), 311-333. | MR 1027058 | Zbl 0687.10028

[6] D.R. Heath-Brown, The Pjateckiĭ-Šapiro prime number theorem. J. Number Theory 16 (1983), 242-266. | MR 698168 | Zbl 0513.10042

[7] M.N. Huxley, M. Nair, Power free values of polynomials III. Proc. London Math. Soc.(3) 41 (1980), 66-82. | MR 579717 | Zbl 0435.10026

[8] H.-Q. Liu, On the number of abelian groups of given order. Acta Arith. 64 (1993), 285-296. | MR 1225430 | Zbl 0790.11074

[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 637350 | Zbl 0462.10029

[10] M. Nair, Power free values of polynomials. Mathematika 23 (1976), 159-183. | MR 429801 | Zbl 0349.10039

[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 513296 | Zbl 0395.10049

[12] F. Roesler, Über die Verteilung der Primzahlen in Folgen der Form [f(n +x)]. Acta Arith. 35 (1979), 117-174. | MR 547671 | Zbl 0342.10027

[13] F. Roesler, Squarefree integers in non-linear sequences. Pacific J. math. 123 (1986), 223-225. | MR 834150 | Zbl 0562.10017

[14] B.R. Srinivasan, Lattice problems of many-dimensional hyperboloids II. Acta Arith. 37 (1963), 173-204. | MR 153641 | Zbl 0118.28201

[15] I.E. Stux, Distribution of squarefree integers in non-linear sequences. Pacific J. Math. 59 (1975), 577-584. | MR 387218 | Zbl 0297.10033

[16] J.D. Vaaler, Some extremal problems in Fourier analysis. Bull. Amer. Math. Soc. 12 (1985), 183-216. | Zbl 0575.42003