The Asymptotic Formula In The Fifth Waring-Goldbach Problem

The Waring-Goldbach problem for k-th powers is to represent all sufficiently large integers satisfying certain congruence conditions as sums of s k-th powers of primes with the number of variables s as small as possible.This is a classical topic since the work of Vinogradov in 1937 and Hua in 1938.In this thesis,we focus on the case k=5.In 2001,Kawada and Wooley showed that all sufficiently large odd numbers n can be written as the sum of 21 fifth powers of prime numbers.In other words,all sufficiently large odd numbers n can be represented as n=p15+p25+…+p215,where p1,…,p21 are prime numbers.The number of variables in the above result is still the best result to date.To prove above result,Kawada and Wooley investigated the number of solutions to the above equation,denoted by R(n),where the variables satisfy certain conditions.By combining the circle method and the sieve method,Kawada and Wooley obtained a lower bound for R(n).The lowered bound obtained by Kawada and Wooley differs from R(n)by a constant.The purpose of this thesis is to study the proof and the method of Kawada and Wooley.Furthermore,by combining the method of Kawada-Wooley and the new estimates for exponential sums over primes established by Kumchev,an asymptotic formula for R(n)is proved.