New Developments in the Additive Theory of Prime Numbers

The Waring-Goldbach problem seeks to represent positive integers satisfying necessary congruence conditions by powers of primes. Ideas using the circle method, sieves, and exponential sums have been incorporated into the Waring-Goldbach problem. This book introduces some of these ideas, illustrated by dealing with the Waring-Goldbach problem.

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The Waring-Goldbach problem seeks to represent positive integers satisfying necessary congruence conditions by powers of primes. The circle method of Hardy and Littlewood in combination with the estimates of Vinogadov for exponential sums over primes gives an affirmative answer to the general Waring-Goldbach problem. In recent years, new ideas using the circle method, sieves, and exponential sums have been incorporated into the Waring-Goldbach problem, encouraging remarkable advances. The purpose of this book is to introduce some of these new ideas, illustrated by dealing with the Waring-Goldbach problem for lower degrees.