Nombres premiers de la forme $\floor{n^c}$

http://dx.doi.org/10.4153/CJM-2001-017-0
Canad. J. Math. 53(2001), 414-433

  Published:2001-04-01
 Printed: Apr 2001

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