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**The Hardy-Littlewood method.
2nd ed.**
*(English)*
Zbl 0868.11046

Cambridge Tracts in Mathematics. 125. Cambridge: Cambridge University Press. vii, 232 p. (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 \(G(k)\) which is defined, for any integer \(k\geq 2\), 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 \(G(k)\)”, 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, \[ G(k)\leq k\log k+k\log\log k+O(k) \quad\text{for large }k, \] 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.

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 \(G(k)\) which is defined, for any integer \(k\geq 2\), 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 \(G(k)\)”, 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, \[ G(k)\leq k\log k+k\log\log k+O(k) \quad\text{for large }k, \] 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.

Reviewer: M.-C.Liu (Hongkong)

### MSC:

11P55 | Applications of the Hardy-Littlewood method |

11-02 | Research exposition (monographs, survey articles) pertaining to number theory |

11P05 | Waring’s problem and variants |