On a Diophantine inequality over primes. II
type: Item
instance of: scholarly article
title: On a Diophantine inequality over primes. II
zbMATH Open document ID: 1129.11045
author: Xiaodong Cao
author: Wenguang Zhai
published in: Monatshefte für Mathematik
publication date: 2007-04-19T00:00:00Z
review text: Suppose that \(c > 1\) is not an integer. Let \(H(c)\) denote the least integer \(r\) such that for any fixed \(\varepsilon > 0\) and for any \(N > N_0(c, \varepsilon) > 0\), the inequality \[ \left| p_1^c + p_2^c + \dots + p_r^c - N \right| < \varepsilon \] has solutions in prime numbers \(p_1, p_2, \dots, p_r\). This analogue of the Waring--Goldbach problem for fractional exponents was first considered by \textit{I. I. Piatetski-Shapiro} [Mat. Sb. 30, 105--120 (1952; Zbl 0047.28001)]. He proved that when \(c \to \infty\), \(H(c) \leq (4 + o(1))c\log c\) and that \(H(c) \leq 5\) in the range \(1 < c < 3/2\). While the former bound is quite sharp and has withstood improvement to this day, the latter has generated a great deal of recent work, including the paper under review. In this paper, the authors prove (a stronger version of) the inequality \(H(c) \leq 5\) for \(1 < c < 81/40\), \(c \neq 2\). What distinguishes this result from earlier improvements on Piatetski-Shapiro's bound is that the range of \(c\) goes beyond \(c = 2\). The main source for this significant improvement is the use of a result of \textit{O. Robert} and \textit{P. Sargos} [J. Reine Angew. Math. 591, 1--20 (2006; Zbl 1165.11067)] on the number of solutions of certain Diophantine inequality.
reviewed by: Angel V. Kumchev
Mathematics Subject Classification ID: 11P32
Mathematics Subject Classification ID: 11D75
zbMATH DE Number: 5145044
zbMATH Keywords: Diophantine inequality
zbMATH Keywords: fractional exponents of primes
zbMATH Keywords: Waring-Goldbach problem
MaRDI profile type: Publication
full work available at URL: s00605-005-0390-4
OpenAlex ID: W1992674904
cites work: On a Diophantine inequality involving prime numbers. III
cites work: On the Waring–Goldbach problem with small non-integer exponent
cites work: SOME RESULTS IN THE ADDITIVE PRIME-NUMBER THEORY
cites work: The Pjateckii-Sapiro prime number theorem
cites work: Three-dimensional exponential sums with monomials
DOI: 10.1007/S00605-005-0390-4
description: scientific article; zbMATH DE number 5145044