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Nombres premiers de la forme ⌊nc⌋

Published online by Cambridge University Press:  20 November 2018

Joël Rivat  and
Patrick Sargos
Joël Rivat
Affiliation:
Institut Élie Cartan, Université Nancy I, B.P. 239, 54506 Vandoeuvre cedex, France, courriel: rivat@iecn.u-nancy.fr
Patrick Sargos
Affiliation:
Institut Élie Cartan, Université Nancy I, B.P. 239, 54506 Vandoeuvre cedex, France, courriel: sargos@iecn.u-nancy.fr
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Abstract

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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 $.

Keywords

11L0711L2011N05
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 53 , Issue 2 , 01 April 2001 , pp. 414 - 433
Copyright
Copyright © Canadian Mathematical Society 2001

References

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