The Waring-Goldbach Problem For Unlike Powers With Almost Equal Variables

As a nonlinear generalization of the Goldbach problem,the Waring-Goldbach problem has always been the focus of research in analytic number theory.The Waring-Goldbach problem mainly studies the possibility that a sufficiently large positive integer N satisfying some congruence conditions can be represented as powers of prime,that is N=p1k+p2k+…+psk,(0-3)where p1,p2,…,pk are primes.In 1937,Vinogradov[1]proved that every sufficiently large odd number can be represented as the sum of three odd primes.In 1966,Chen[2]proved that every sufficiently large even number can be represented as the sum of an even prime and an odd number with at most two prime factors.On the other hand,many researchers have been attracted to study the WaringGoldbach problem under appropriate conditions.The general approach is to limit the variables to short intervals,this kind of problem is called Waring-Goldbach problem with almost equal variables.Specifically,the problem studies the solvability of the following equations:where 1≤i≤s,U=N1-θk,s,θk,s∈(0,1).The exceptional sets of this problem has also attracted the attention of researchers.Let Ek,s(N,U)be the number of integers up to N satisfying some conditions that can not be represented at the form(0-4).The problem aims to prove for suitable 0<θk,s<1,Ek,s(N,U)(?)N1-θk,s-ε.When k=3,s=5,in 2012,Liu and Sun[4]proved that θ3,5=1/102.In the same year,Li[5]improved the value of θ3.5=1/48.In 2015,by using the new method of Zhao[18]in minor arcs,Ren and Yao[6]to treat the θ3,5=7/261.In 2019,Wang[7]proved that θ3,5=1/27,this is the best result at present.In this paper,we study the generalization of this problem.That is,replace one cube of prime by a k(k≥4)power of prime,where 1≤i≤4,U=N1-θ,i=1,2,3,4,k≥4.We define I(N,5U)=[N-5U,N+5U],let E(N,U)denote the number of n∈I(N,5U)satisfying some congruent conditions that can not be represented as(0-5).We haveTheorem Let k≥4,E(N,U)(?)U1-ε holds forThe basic method to solve the Waring-Goldbach problem is the circle method,which divides interval[0,1]into the major arcs and minor arcs,and then estimates the integral on the major arcs and the minor arcs respectively.We will use this method in this paper.For the major arcs,we will use the idea of increasing the major arcs proposed by Liu[8]and the iterative method established by Liu[9]in another paper which are mainly used to process the major arcs.When dealing with the minor arcs,according to the main parameters P*,Q*choose in using the circle method,we separate two cases 4≤k≤6 and k≥7 in proof.When 4≤k≤6,we will use the idea of exponential sum estimation by Zhao[18],and the mean value estimation by Yao[20];when k≥7,we use value estimation by Hua’s[21]and Kumchev[22]lemma which play a vital role in the proof.