Nombres premiers de la forme ⌊nc ⌋

Rivat, Joël ; Sargos, Patrick

Canadian journal of mathematics, Tome 53 (2001) no. 2, pp. 414-433

Résumé

For $c\,>\,1$ we denote by ${{\pi }_{c}}\left( x \right)$ the number of integers $n\,\le \,x$ such that $\left\lfloor {{n}^{c}} \right\rfloor $ is prime. In 1953, Piatetski-Shapiro has proved that ${{\pi }_{c}}\left( x \right)\,\sim \,\frac{x}{c\,\log \,x},\,x\to \,+\infty $ holds for $c\,<\,12/11$ . Many authors have extended this range, which measures our progress in exponential sums techniques. In this article we obtain $c\,<\,1.16117\ldots $ .

Détail

DOI : 10.4153/CJM-2001-017-0

Mots-clés : 11L07, 11L20, 11N05

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