Additive Theory of Prime Numbers

Examines additive number theory, paying particular attention to the work of Loo-Keng Hua, best remembered for his contributions to Waring's Problem and his estimates of trigonometric sums. This is an exposition of the classic methods as well as Hua's own techniques, many of which have now also become classic.

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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. This is a reprint of the 1965 original. (MMONO/13.2)