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.