Exceptional Sets In Waring-Goldbach Problem For Sixth Powers

As a generalization of Goldbach’s problem,Waring-Goldbach problem has always been the focus of analytic number theory.The Waring-Goldbach problem is to study whether a positive integer that satisfies the congruence conditions can be expressed as the sum of prime powers.That is,for sufficiently large positive integers n,whether there are prime numbers p1,p2,…,ps,such that n=p1k+p2k+…+psk.The philosophy thought contained in the Hardy-Littlewood circle method leads number theorists to believe that for k and s,two natural numbers satisfying s>k+1,there exist a fixed modulus qk,s and a set Nk,s of congruence classes mod qk,s such that for all large enough n∈Nk,s can be expressed as the form(0-1).Although it is not yet possible to prove this conjecture,mathematicians have achieved a lot of good results on the Waring-Goldbach problem.For a long time,mathematicians have been studying the upper bound of the H(k).Define H(k)to be the smallest s,for sufficiently large n∈Nk,s can write(0-1).1938,Hua[1]proved that H(k)≤2k+1 is true for k≥1.For k ≤3,this is still the best result so far.For k≥4,Zhao[2],Kawada and Wooley[3],Kumchev and Wooley[4]have continuously improved the upper bound of H(k).Later,number theorists found that they could further reduce the number of variables which are needed to solve(0-1)by replacing all sufficiently large n with almost all sufficiently large n.Naturally,number theorists began to study the size of the exception set Ek,s(N).Let Ek,s(N)be the number of integers n≤N satisfying n≡s(mod qk,s),that(??)cannot be solved in primes p1,p2,…,ps.Mathematicians mainly estimate the upper bound of Ek,s(N).For example,Kawada and Wooley[5]geted the nontrivial upper bound of for 4 ≤s ≤6 by the relations between exception sets.For the case of k=5 and k=6,the nontrivial upper bound of Ek,s(N)is given by Kumchev[6].The latest results of E6,s(N)came from Feng[7].In this article,we mainly study the Ek,s(N)for k＝6 and 16 ≤s ≤31.In this case,we set qk,s=504.For the case of 16 ≤s ≤19,this paper adopts the classical circle method to solve the problem.First,the solution of the equation is converted into an integral on[0.1],and then we decompose the unit interval into sets of major and minor arcs.The integral on sets of major and minor arcs are estimated separately.For the integral on set of major arc.Theorem 1.2.1 in Liu[8]is applied in this paper to give its lower bound.For the integral on set of minor arc,it is further divided into two parts.In one part,Theorem 1.2.1 in Ren[9]and Lemma 6.3 in Zhao[2]are used to get the upper bound.In the other part,we will try to use Lemma 2.2.2 in Zhao[2]and Lemma 2.2.1 and Lemma 2.2.2 in Kumchev and Wooley[3]to get the upper bound,and finally get the exception set of this problem.For the case of 20 ≤s ≤31,we use the methods provided by Kawada and Wooley[5]to create relationships between exception sets and get the results.This paper improves the result of Feng[7]to get a better upper bound of E6,s(N)for s=18 and 20 ≤s ≤31,which is Theorem 1.2.1.