# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
The Hardy-Littlewood method. 2nd ed. (English) Zbl 0868.11046
Cambridge Tracts in Mathematics. 125. Cambridge: Cambridge University Press. vii, 232 p. £35.00; \$ 49.95 (1997).

The first edition of the book was published about 16 years ago and was reviewed very soon after publication. For the review articles, see, for example, A. Baker [Bull. Lond. Math. Soc. 14, 362-364 (1982)], W. Schmidt [Bull. Am. Math. Soc. 7, 434-437 (1982)] and the reviewer [1981; Zbl 0455.10034 and 1984; MR 84b:10002]. The reviewer would like to write the present review as a supplement to his review of the first edition since the essential difference between the two editions is only the addition of Wooley’s work, which is included as a new chapter in the second edition.

Since the publication of the first edition there has been a series of important developments in the areas concerning the Hardy-Littlewood method. In particular, the recent results by Wooley and the author have led to considerable improvements on some central problems of the Waring problem, namely, estimation of upper bounds for $G\left(k\right)$ which is defined, for any integer $k\ge 2$, to be the least integer $s$ such that every sufficiently large natural number is a sum of at most $s$ $k$-th powers of natural numbers.

In order to include these new developments the author adds a new chapter, i.e., Chapter 12, “Wooley’s upper bound for $G\left(k\right)$”, and a new section to Chapter 5, i.e., Section 5.5, “Wooley’s refinement of Vinogradov’s mean”, in the second edition of his book. In Chapter 12, a detailed proof for the result,

$G\left(k\right)\le klogk+kloglogk+O\left(k\right)\phantom{\rule{1.em}{0ex}}\text{for}\phantom{\rule{4.pt}{0ex}}\text{large}\phantom{\rule{4.pt}{0ex}}k,$

is provided.

Correspondingly, new exercises and many titles of recent papers have been added to some chapters and the bibliography of the book. Besides these new materials, in the second edition the author maintains all the original contents appearing in the first edition.

The book is well written and self-contained. However, proofs of the theorems in the book are quite condensed and they require a certain degree of analytic skill to understand the details. The book will continue to be of interest for both specialists and research students, and to be a well-known standard reference of the Hardy-Littlewood method in research for many years to come.

##### MSC:
 11P55 Applications of the Hardy-Littlewood method 11-02 Research monographs (number theory) 11P05 Waring’s problem and variants