Waring-Goldbach problem for unlike powers
type: Item
instance of: scholarly article
title: Waring-Goldbach problem for unlike powers
zbMATH Open document ID: 1128.11043
author: Xiumin Ren
author: Kai-Man Tsang
publication date: 2007-06-07T00:00:00Z
review text: The problem considered here is that of representing a positive integer~\(n\) as a sum \(p_2^ 2+p_3^ 3+p_4^ 4+p_5^ 5\) of prime powers. Let \(E(N)\) denote the number of ``exceptional'' even numbers \(n \leq N\) that are not expressible in this way. \textit{K.~Prachar} [Monatsh. Math. 57, 66--74 (1953; Zbl 0050.04003)] proved that \(E(N) \ll N/\log^ cN\) whenever \(c < {30 \over 47}\), and more recently \textit{C.~Bauer} [Rocky Mt. J. Math. 1, No. 4, 1151--1170 (2001; Zbl 1035.11047)] showed that \(E(N) \ll N^ {1-\delta}\) for some certain \(\delta\) slightly better than \(1\over2742\). The authors now establish the result with \(\delta={1\over66}\). In the problem in which the first power~\(p_1\) of a prime~\(p_1\) is added to the summand, \textit{K. Prachar} [Monatsh.\ Math.~57, 113--116 (1953; Zbl 0052.27901)] proved that all sufficiently large odd integers are representable, a fact which the authors recover as a corollary of their main result. The proof uses the circle method, using a approach used by \textit{J. Y. Liu} and \textit{T. Zhan} in [Sci. China, Ser. A 41, No. 7, 710--722 (1998; Zbl 0938.11048)], and with \textit{M. C. Liu} in [Monatsh. Math. 128, No. 4, 283--313 (1999; Zbl 0940.11047)], which allows enlarged major arcs in a way that does not invoke the Deuring--Heilbronn phenomenon or an appeal to the properties of a possible Siegel zero. This approach appears in a number of other papers in the recent literature, one of them [Q. J. Math. 52, No. 1, 107--126 (2001; Zbl 0991.11056)] by the first-named author of the paper currently under review. The authors are now able to make use of classical zero-density estimates for \(L\)-functions, in a way that they make more efficient by a certain iterative treatment. On the minor arcs, that authors use an estimate of \textit{K. Kawada} and \textit{T. D. Wooley} from a paper [Proc. Lond. Math. Soc. (3) 83, No. 1, 1--50 (2001; Zbl 1016.11046)] on the Waring-Goldbach problem for fourth and fifth powers. They also u…
Mathematics Subject Classification ID: 11P32
Mathematics Subject Classification ID: 11P05
Mathematics Subject Classification ID: 11P55
zbMATH DE Number: 5162303
zbMATH Keywords: circle method
zbMATH Keywords: exceptional set
zbMATH Keywords: enlarged major arcs
reviewed by: George Greaves
MaRDI profile type: Publication
full work available at URL: s10114-005-0733-z
OpenAlex ID: W1984649548
cites work: A Problem in Additive Number Theory
cites work: Über ein Problem vom Waring-Goldbachschen Typ
cites work: Über ein Problem vom Waring-Goldbachschen Typ. II
cites work: An improvement on a theorem of the Goldbach-Waring type.
cites work: Sums of five almost equal prime squares. II
cites work: Squares of primes and powers of 2
cites work: The exceptional set in Roth's theorem concerning a cube and three cubes of primes
cites work: On the Waring-Goldbach problem for fourth and fifth powers
cites work: Q5830713
cites work: Q3900124
cites work: Large values of Dirichlet polynomials, III
cites work: On Linnik's constant.
cites work: Q3735790
DOI: 10.1007/S10114-005-0733-Z
published in: Acta Mathematica Sinica, English Series
description: scientific article; zbMATH DE number 5162303