Lectures Series on the Circle Method
and its application to Waring’s problem

Y.K. Lau, X. Ren, K.M. Tsang
The University of Hong Kong



The circle method is an important and powerful tool in analytic number theory. It is particularly useful in handling additive equations. Together with techniques from other areas, such as the sieve theory and the theory of zeta and L-functions, etc., it enables us to obtain very interesting results on problems of representation of integers by sums of integers of special types.

The aim of this series of lectures is to introduce the circle method and some of its related techniques, with the ultimate goal of proving some interesting results of the Waring-Goldbach type. The lectures are arranged into the following five parts.

A.    Waring’s problem and basics of the circle method.
B.    Preliminary estimates of exponential sums and certain mean values.
C.    The Vinogradov three-prime theorem.
D.    Waring-Goldbach problems for low degrees.
E.    Further techniques and recent developments.

References:
1.    H. Davenport, Multiplicative Number Theory, GTM 74, Springer-Verlag, 1980.
2.    L.K. Hua, Additive Theory of Prime Numbers, AMS, Rhode Island, 1966.
3.    C.D. Pan and C.B. Pan, Goldbach Conjecture, Science Press, Beijing, 1992.
4.    E.C. Titchmarsh (revised by D.R. Heath-Brown), The Theory of the Riemann Zeta-function, Oxford University Press, 1988.
5.    R.C. Vaughan, The Hardy-Littlewood Method, 2nd Ed., Cambridge University Press, 1990.