**Translations of Mathematical Monographs**

1965;
190 pp;
Softcover

MSC: Primary 11;

**Print ISBN: 978-0-8218-4942-2
Product Code: MMONO/13.S**

List Price: $75.00

AMS Member Price: $60.00

MAA Member Price: $67.50

**Electronic ISBN: 978-1-4704-1621-8
Product Code: MMONO/13.S.E**

List Price: $70.00

AMS Member Price: $56.00

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# Additive Theory of Prime Numbers

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*L. K. Hua*

Loo-Keng Hua was a master mathematician, best known for his work using analytic methods in number theory. In particular, Hua is remembered for his contributions to Waring's Problem and his estimates of trigonometric sums. Additive Theory of Prime Numbers is an exposition of the classic methods as well as Hua's own techniques, many of which have now also become classic. An essential starting point is Vinogradov's mean-value theorem for trigonometric sums, which Hua usefully rephrases and improves. Hua states a generalized version of the Waring-Goldbach problem and gives asymptotic formulas for the number of solutions in Waring's Problem when the monomial \(x^k\) is replaced by an arbitrary polynomial of degree \(k\). The book is an excellent entry point for readers interested in additive number theory. It will also be of value to those interested in the development of the now classic methods of the subject.

#### Table of Contents

# Table of Contents

## Additive Theory of Prime Numbers

- Cover Cover11
- Title page i2
- Contents iii4
- Foreword vii8
- Preface to the present edition ix10
- Preface to the 1953 Chinese edition ix10
- Preface to the Russian edition x11
- Preface originally intended for the Russian edition xi12
- Explanatory remarks xiii14
- Trigonometric sums 116
- Estimates for sums involving the divisor function 𝑑(𝑛) 1126
- Mean-value theorems for certain trigonometric sums (I) 1934
- Vinogradov’s mean-value theorem and its corollaries 2641
- Mean-value theorems for certain trigonometric sums (II) 4055
- Trigonometric sums depending on prime numbers 6681
- An asymptotic formula for the number of solutions of the Waring-Goldbach problem 7893
- Singular series 100115
- A further study of the Waring-Goldbach problem 109124
- Indeterminate equations in prime unknowns 124139
- A further study of the problem of the preceding chapter 157172
- Other results 175190
- Appendix 181196
- Back Cover Back Cover1206