On the Piatetski-Shapiro-Vinogradov theorem

Angel Kumchev

Angel Kumchev

Journal de théorie des nombres de Bordeaux, Tome 9 (1997) no. 1, pp. 11-23.

Résumé
Abstract

Dans cet article, nous considérons la formule asymptotique pour le nombre de représentations d’un entier impair $N$ sous la forme $p_{1} + p_{2} + p_{3} = N$ , où les $p_{i}$ sont des nombres premiers du type $p_{i} = [n^{1 / γ_{i}}]$ ; nous utilisons la méthode de van der Corput en dimension deux et nous étendons le domaine de validité de la formule asymptotique en affaiblissant les hypothèses sur les $γ_{i}$ . Dans le cas le plus intéressant $γ_{1} = γ_{2} = γ_{3} = γ$ , notre résultat entraîne que tout entier impair assez grand s’écrit comme la somme de trois nombres premiers de Piatetski-Shapiro du type $γ$ pour $50 / 53 < γ < 1$ .

In this paper we consider the asymptotic formula for the number of the solutions of the equation $p_{1} + p_{2} + p_{3} = N$ where $N$ is an odd integer and the unknowns $p_{i}$ are prime numbers of the form $p_{i} = [n^{1 / γ_{i}}]$ . We use the two-dimensional van der Corput’s method to prove it under less restrictive conditions than before. In the most interesting case $γ_{1} = γ_{2} = γ_{3} = γ$ our theorem implies that every sufficiently large odd integer $N$ may be written as the sum of three Piatetski-Shapiro primes of type $γ$ for $50 / 53$ < $γ$ < $1$ .

EuDML MR Zbl

Mots clés : Piatetski-Shapiro primes, Goldbach problem, exponential sums

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     title = {On the {Piatetski-Shapiro-Vinogradov} theorem},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {11--23},
     publisher = {Universit\'e Bordeaux I},
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     number = {1},
     year = {1997},
     zbl = {0890.11029},
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Angel Kumchev. On the Piatetski-Shapiro-Vinogradov theorem. Journal de théorie des nombres de Bordeaux, Tome 9 (1997) no. 1, pp. 11-23. https://jtnb.centre-mersenne.org/item/JTNB_1997__9_1_11_0/

Bibliographie
Cité par

[1] R.C. Baker, G. Harman, J. Rivat, Primes of the form [nc], J. Number Theory 50 (1995), 261-277. | MR | Zbl

[2] A. Balog, J.P. Friedlander, A hybrid of theorems of Vinogradov and Piatetski-Shapiro, Pasific J. Math. 156 (1992), 45-62. | MR | Zbl

[3] S.W. Graham, G.A. Kolesnik, Van der Corput's Method of Exponential Sums, L.M.S. Lecture Notes 126, Cambridge University Press, 1991. | Zbl

[4] D.R. Heath-Brown, The Piatetski-Shapiro prime number theorem, J. Number Theory 16 (1983), 242-266. | MR | Zbl

[5] C.-H. Jia, On the Piatetski-Shapiro prime number theorem (II), Science in China Ser. A 36 (1993), 913-926. | MR | Zbl

[6] ____, On the Piatetski-Shapiro prime number theorem, Chinese Ann. Math. 15B:1 (1994), 9-22. | Zbl

[7] ____, On the Piatetski-Shapiro-Vinogradov theorem, Acta Arith. 73 (1995), 1-28. | MR | Zbl

[8] G.A. Kolesnik, The distribution of primes in sequences of the form [nc], Mat. Zametki 2 (1972), 117-128. | MR | Zbl

[9] ____, Primes of the form [nc], Pacific J. Math. 118 (1985), 437-447. | MR | Zbl

[10] A. Kumchev, On the distribution of prime numbers of the form [nc], (preprint). | MR

[11] D. Leitmann, Abschatzung trigonometrischer summen,, J. Reine Angew. Math. 317 (1980), 209-219. | MR | Zbl

[12] H.-Q. Liu, J. Rivat, On the Piatetski-Shapiro prime number theorem, Bull. London Math. Soc. 24 (1992), 143-147. | Zbl

[13] I.I. Piatetski-Shapiro, On the distribution of prime numbers in sequences of the form [f(n)], Mat. Sb. 33 (1953), 559-566. | MR | Zbl

[14] J. Rivat, Autour d'un theoreme de Piatetski-Shapiro, Thesis, Université de Paris Sud, 1992.

[15] I.M. Vinogradov, Representation of an odd number as the sum of three primes, Dokl. Akad. Nauk. SSSR 15 (1937), 291-294. | JFM | Zbl

[16] E. Wirsing, Thin subbases, Analysis 6 (1986), 285-306. | MR | Zbl