The Exceptional Set In Mixed Waring-Goldbach Problem

The aring-Goldbach problem has drown a lot of attention as an impor-tant part of additive theory of prime numbers in the long run.It seeks to find a positive integer s as small as possible,such that every positive integer n satisfying necessary congruence can be represented as where s is dependent on k and p1,p2,...,ps are primes.When k = 1,s = 3,it's the Odd Goldbach conjecture(every big enough positive odd integer can be represented by sums of three primes),which is also called the Three prime number theorem and was proved by Vinogradov[1]in 1937 with the help of analytic methods.When k=1,s = 2,its's the Even Goldbach conjecture which has not been proved so far.Generally,We are more interested in the continuous power Waring-Goldbach problem with lower powers,because it tends to achieve relatively better results.For instances:for k= 3,in 1937,Hua[2]proved that every big enough odd n can be represented by sums of nine cubes of prime with at most E(N)?Nlog-AN exceptions,all positive integers n<N satisfying some necessary congruence conditions,where A>0 is arbitrary.As time went by,this result has been greatly improved,and meanwhile,people come into being interested in the mixed powers Waring problem.For example,an interesting question is to represent a positive integers as the sum of one square,four cubes,one b power and one c power of nature numbers.Let Rb,c(n)denote the number of the representations.In 1981,Hooley[3]first got an asymptotic formula for R3,5(n).From Brudern's work[4],one can easily get the formula for R3,c(n).The main work of this paper is to solve a question of the mixed power Waring-Goldbach problem using the method in studying the continuous powerWaring-Goldbach problem.It is proved that with at most O(N1-21-k/k+?)exceptions,all positive integers n,n ? N and N<n ? 2N,satisfying some necessary congruence conditions can be written as sums of four prime cubes and a k power of a prime,for any fixed k and k?3(Theorem 1.1).However,there are very limited progresses on the question of n = P13p23?p33?p43 but Ren[5]has proved that there is a positive density for n which can be represented by four prime cubes.This guaranteed that the question studied in this paper can achieve a relatively better result.In order to get the estimate for the exceptional set,we need to make the major arcs as large as possible with the help of circle methods.We apply the approach of Liu[6]to treat the enlarged major arcs in the Waring-goldbach problem.We can take advantage of the iterative method established by Liu[7]in the circle method to estimate the singular series and finally get a lower bound of the major arcs.To handle the minor arcs,we make use of the method of Zhao[8],separating the minor arcs into two parts.With the help of estimates of exponential sums over primes established by Ren[9]and Zhao[8],and mean-value estimates,we can finally get an upper bound of the minor arcs,and further more,the exceptional set of the question in this paper.