On the Piatetski-Shapiro-Vinogradov theorem
type: Item
instance of: scholarly article
title: On the Piatetski-Shapiro-Vinogradov theorem
zbMATH Open document ID: 0890.11029
published in: Journal de Théorie des Nombres de Bordeaux
publication date: 1998-03-25T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/93674
full work available at URL: jtnb9-1.html
review text: Let \(\gamma\in (1,{53 \over 50})\) and let \({\mathcal P} \equiv {\mathcal P} (\gamma)= \{p:p= [n^\gamma]\), \(n\in\mathbb{N}\}\) be the set of Piatetski-Shapiro primes. Define \(T(N) \equiv T(\gamma_1, \gamma_2, \gamma_3;N) =|\{P_i\in {\mathcal P} (\gamma_i): P_1+ P_2+ P_3= N\}|\). The author proves that if \(73(1-\gamma_3) <9\), \(73(1-\gamma_2) +43(1-\gamma_3) <9\) and \(73(1-\gamma_1) +43(2-\gamma_2 -\gamma_3) <9\), then \[ T(N)= \bigl(1 +o(1)\bigr) {\Gamma (\gamma_1+1) \Gamma(\gamma_2+1) \Gamma(\gamma_3+1) \over \Gamma (\gamma_1+ \gamma_2+ \gamma_3)} \sigma(N) N^{\gamma_1 +\gamma_2+ \gamma_3-1}/ \log^3N, \] where \[ \sigma(N) =\prod_{p\mid N} \left(1- {1\over (p-1)^2} \right) \prod_{p \nmid N} \left(1+ {1\over (p-1)^3} \right) \] is the singular series from Vinogradov's result on the ternary Goldbach problem.
Mathematics Subject Classification ID: 11P32
zbMATH DE Number: 1132884
zbMATH Keywords: Piatetski-Shapiro-Vinogradov theorem
zbMATH Keywords: Piatetski-Shapiro primes
zbMATH Keywords: ternary Goldbach problem
author: Angel V. Kumchev
reviewed by: G. A. Kolesnik
MaRDI profile type: Publication
OpenAlex ID: W2043044756
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cites work: THIN SUBBASES
DOI: 10.5802/JTNB.186
description: scientific article; zbMATH DE number 1132884