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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, 11L20, 11N05 show english descriptions Estimates on exponential sums
Sums over primes
Distribution of primes
11L07 - Estimates on exponential sums
11L20 - Sums over primes
11N05 - Distribution of primes
 

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