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

Xiaodong Cao; Wenguang Zhai

doi:10.5802/jtnb.229

The distribution of square-free numbers of the form

[n^{c}]

Xiaodong Cao ; Wenguang Zhai

Journal de Théorie des Nombres de Bordeaux, Tome 10 (1998) no. 2, pp. 287-299.

Résumé
Abstract

Nous montrons que pour $1 < c < \frac{61}{36} = 1.6944 \dots,$ la suite $[n^{c}] (n = 1, 2, \dots)$ 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 $[n^{c}] (n = 1, 2, \dots)$ contains infinite squarefree integers whenever $1 < c < \frac{61}{36} = 1.6944 \dots$ , which improves Rieger’s earlier range $1 < c < 1.5$ .

MR 1828246 | Zbl 0926.11066

DOI : https://doi.org/10.5802/jtnb.229
Mots clés : square-free number, exponential sum, exponent pair

@article{JTNB_1998__10_2_287_0,
     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 = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.229/}
}

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/articles/10.5802/jtnb.229/

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