# On the Piatetski-Shapiro-Vinogradov theorem

• Volume: 9, Issue: 1, page 11-23
• ISSN: 1246-7405

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## Abstract

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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 ${p}_{i}=\left[{n}^{1/{\gamma }_{i}}\right]$. We use the two-dimensional van der Corput’s method to prove it under less restrictive conditions than before. In the most interesting case ${\gamma }_{1}={\gamma }_{2}={\gamma }_{3}=\gamma$ our theorem implies that every sufficiently large odd integer $N$ may be written as the sum of three Piatetski-Shapiro primes of type $\gamma$ for $50/53$ &lt; $\gamma$ &lt; $1$.

## How to cite

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Kumchev, Angel. "On the Piatetski-Shapiro-Vinogradov theorem." Journal de théorie des nombres de Bordeaux 9.1 (1997): 11-23. <http://eudml.org/doc/248000>.

@article{Kumchev1997,
abstract = {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 $p_i = [ n^\{1/ \gamma _i\}]$. We use the two-dimensional van der Corput’s method to prove it under less restrictive conditions than before. In the most interesting case $\gamma _1 = \gamma _2 = \gamma _3 = \gamma$ our theorem implies that every sufficiently large odd integer $N$ may be written as the sum of three Piatetski-Shapiro primes of type $\gamma$ for $50/53$ &lt; $\gamma$ &lt; $1$.},
author = {Kumchev, Angel},
journal = {Journal de théorie des nombres de Bordeaux},
keywords = {Piatetski-Shapiro primes; Goldbach problem; exponential sums; Piatetski-Shapiro-Vinogradov theorem; ternary Goldbach problem},
language = {eng},
number = {1},
pages = {11-23},
publisher = {Université Bordeaux I},
title = {On the Piatetski-Shapiro-Vinogradov theorem},
url = {http://eudml.org/doc/248000},
volume = {9},
year = {1997},
}

TY - JOUR
AU - Kumchev, Angel
TI - On the Piatetski-Shapiro-Vinogradov theorem
JO - Journal de théorie des nombres de Bordeaux
PY - 1997
PB - Université Bordeaux I
VL - 9
IS - 1
SP - 11
EP - 23
AB - 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 $p_i = [ n^{1/ \gamma _i}]$. We use the two-dimensional van der Corput’s method to prove it under less restrictive conditions than before. In the most interesting case $\gamma _1 = \gamma _2 = \gamma _3 = \gamma$ our theorem implies that every sufficiently large odd integer $N$ may be written as the sum of three Piatetski-Shapiro primes of type $\gamma$ for $50/53$ &lt; $\gamma$ &lt; $1$.
LA - eng
KW - Piatetski-Shapiro primes; Goldbach problem; exponential sums; Piatetski-Shapiro-Vinogradov theorem; ternary Goldbach problem
UR - http://eudml.org/doc/248000
ER -

## References

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11. [11] D. Leitmann, Abschatzung trigonometrischer summen,, J. Reine Angew. Math.317 (1980), 209-219. Zbl0421.10024MR581343
12. [12] H.-Q. Liu, J. Rivat, On the Piatetski-Shapiro prime number theorem, Bull. London Math. Soc.24 (1992), 143-147. Zbl0772.11032
13. [13] I.I. Piatetski-Shapiro, On the distribution of prime numbers in sequences of the form [f(n)], Mat. Sb.33 (1953), 559-566. Zbl0053.02702MR59302
14. [14] J. Rivat, Autour d'un theoreme de Piatetski-Shapiro, Thesis, Université de Paris Sud, 1992.
15. [15] I.M. Vinogradov, Representation of an odd number as the sum of three primes, Dokl. Akad. Nauk. SSSR15 (1937), 291-294. Zbl0016.29101JFM63.0131.04
16. [16] E. Wirsing, Thin subbases, Analysis6 (1986), 285-306. Zbl0586.10032MR832752

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