The Waring-Goldbach Problem For Unlike Powers

The Waring-Goldbach problem has always been one of the classical topics in addictive prime number theory,and is of great significance.The Waxing-Goldbach problem studies the solvability of the equation n=p1k+p2k+…+psk where n satisfies some necessary congruence conditions,k is a positive integer and p1,p2,…,ps are primes.Denote by H(k)the minimum of s for which the above equation is solvable.The Goldbach Conjecture states that H(1)=2.In 1937,Vinogradov[2]proved that any sufficiently large odd number can be expressed as the sum of three primes by using the circle method and the estimation of exponential sum over primes,which is the three primes theorem,i.e.H(1)=3.In 2013,Helfgott[3,4]solved the problem completely.After that,a series of results for different values of k were obtained.For example,Zhao[11]proved that H(4)≤13.It is conjectured that H(k)=k+1,but this is far out of reach at present for k≥1.We usually consider exceptional sets for the problem when s is small.Denote by Ek,s(N)the number of n up to N satisfying some necessary conditions that can not be represented as the above equation.For k=4 and s=9,Hua’s method of[14]led to the following result that E4,9(N)<<N(log N)-A.After that,lots of improvements were obtained.In 2005,Kumchev[15]proved that(?) In 2019,Feng and Liu[16]proved that (?)This is the best result so far.Considering the generalization of the above problem,we study the exceptional set of the Waring-Goldbach problem for unlike powers of the form(?) Let N denote the set of all positive integers n satisfying necessary congruence conditions and Ek(N)denote the number of positive integers n∈N and N/2<n≤N that cannot be written as sums of eight fourth powers of primes and a kth powers of a prime.The result of this paper is as follows.(?)where(?)In order to prove this result,we mainly use the circle method founded by Hardy and Littlewood.This paper is divided into four chapters.In the first chapter,we introduce the research background and the main result.In the second chapter,we briefly introduce the general idea of the proof.In this chapter,we divide the interval[0,1]into major arcs and minor arcs by using Dirichlet rational approximation theorem,and then estimate the integral on the major arcs and minor arcs respectively.In the third chapter,we will deal with the integral on the major arcs.We mainly use the method of expanding the major arcs proposed by Liu and Zhan in[18].In the fourth chapter,we will deal with the integral on the minor arcs.In order to obtain the estimation of the integral,we divide the minor arcs into two parts and use the estimation of exponential sum over primes as well as new bound of integral moments.Moreover we will borrow the method of Zhao[11].