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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: [email protected]
Patrick Sargos
Affiliation:
Institut Élie Cartan, Université Nancy I, B.P. 239, 54506 Vandoeuvre cedex, France, courriel: [email protected]
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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

Références

[1] Baker, R. C., Harman, G., and Rivat, J., Primes of the form [nc]. J. Number Theory 50 (1995), 261–277.Google Scholar
[2] Deshouillers, J.-M., Nombres premiers de la forme [nc]. C. R. Acad. Sci. Paris Sér. A–B 282(1976), A131–A133.Google Scholar
[3] Fouvry, E. and Iwaniec, H., Exponential Sums with Monomials. J. Number Theory 33 (1989), 311–333.Google Scholar
[4] Friedlander, J. and Iwaniec, H., Using a parity-sensitive sieve to count prime values of a polynomial. Proc. Nat. Acad. Sci. USA 94 (1997), 1054–1058.Google Scholar
[5] Graham, S. and Kolesnik, G., Van der corput’s Method of Exponential Sums. London Math. Soc. Lecture Note Ser. 126, Cambridge University Press, 1991.Google Scholar
[6] Heath-Brown, D. R., Prime numbers in short intervals and a generalized Vaughan identity. Canad. J. Math. 34 (1982), 1365–1377.Google Scholar
[7] Heath-Brown, D. R., The Piatetski-Shapiro Prime Number Theorem. J. Number Theory 16 (1983), 242–266.Google Scholar
[8] Huxley, M., Area, Lattice Points and Exponential Sums. London Math. Soc. Monographs New Series 13, Oxford Science Publications, 1996.Google Scholar
[9] Jia, C.-H., On Piatetski-Shapiro Prime Number Theorem II. Science in China 36 (1993), 913–926.Google Scholar
[10] Kolesnik, G. A., The distribution of primes in sequences of the form [nc]. Mat. Zametki 2 (1967), 117–128.Google Scholar
[11] Kolesnik, G. A., Primes of the form [nc]. Pacific J. Math. 118 (1985), 437–447.Google Scholar
[12] Leitmann, D. and Wolke, D., Primzahlen der Gestalt [f(n)]. Math. Z. 145 (1975), 81–92.Google Scholar
[13] Liu, H.-Q. and Rivat, J., On the Piatetski-Shapiro Prime Number Theorem. Bull. London Math. Soc. 24 (1992), 143–147.Google Scholar
[14] Piatetski-Shapiro, I., On the distribution of prime numbers in sequences of the form [f(n)]. Math. Sbornik 33 (1953), 559–566.Google Scholar
[15] Rivat, J., Autour d’un théorème de Piatetski-Shapiro (Nombres premiers dans la suite [nc]). Thèse de Doctorat, Université de Paris-Sud, 1992.Google Scholar
[16] Sargos, P., Un critère de la dérivée cinquième pour les sommes d’exponentielles. Bull. LondonMath. Soc. 32 (2000), 398–402.Google Scholar
[17] Titchmarsh, E. C., The Theory of the Riemann Zeta-function, revised by D. R. Heath-Brown. Oxford Science Publications, 2nd edition, 1986.Google Scholar
[18] Vaaler, J., Some extremal functions in Fourier analysis. Bull. Amer.Math. Soc. 12 (1985), 183–216.Google Scholar