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

Published:2001-04-01
Printed: Apr 2001
• Joël Rivat
• Patrick Sargos
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## Abstract

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 $c < 1.16117\dots\;$.
 MSC Classifications: 11L07 - Estimates on exponential sums 11L20 - Sums over primes 11N05 - Distribution of primes