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, Volume 10 (1998) no. 2, pp. 287-299.

Abstract
Résumé

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 $1 < c < 1.5$ .

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 $1 < c < 1.5 .$

MR: 1828246 | Zbl: 0926.11066

DOI: 10.5802/jtnb.229
Keywords: square-free number, exponential sum, exponent pair

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     title = {The distribution of square-free numbers of the form $[n^c]$},
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     publisher = {Universit\'e Bordeaux I},
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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, Volume 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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