On The Waring-Goldbach Problem With Almost Equal Summands

Let n be a sufficiently large positive integer subject to certain local congruent conditions and κ be a positive integer.The Waring-Goldbach problem studies the representation of n as a sum of powers of primes,i.e.n = p1κ+P2κ+ …+Psκ,(0.3)where p1,…,ps are prime numbers.If we take k = 1,s = 2 in the equation(0.3),then it becomes the famous Goldbach Conjecture(the binary Goldbach Con-jecture).Namely,the Waring-Goldbach problem is a nonlinear extension of the Goldbach Conjecture.For the linear case of the Waring-Goldbach problem,in 1937 Vinogradov[44]showed that when s≥3,the equation(0.3)has prime solutions for any sufficiently large odd integer n,which is called the famous Vinogradov’s Three Primes The-orem.In 2013,Helfgott[9,10]proved that when s≥ 3,for all odd integers n ≥ 9,the equation(0.3)has prime solutions,which entirely solves the ternary Goldbach Conjecture.For the nonlinear case of the Waring-Goldbach problem,in 1938 Hua[11]first proved that when s ≥2κ + 1,the equation(0.3)has prime so-lutions for any integer k ≥ 1,which was systematically summarized in[12].This result is still the best when k ≤3.When κ≥ 4,many scholars improved this result(see[15,16,18,19,39,40,42,49]).Another interesting problem in Number Theory is the Waring-Goldbach prob-lem with almost equal summands.Now,we introduce this problem in detail.First let τ = τ(k,p)be the largest integer satisfying pτ| k,and define(?)(?)(?)(?)We study the equation(0.3)for n restricted to the congruence class(?)Given a sufficiently large integer n∈ Hk,s,the Waring-Goldbach problem with almost equal summands studies whether the equation(0.3)have prime solutions such that(?)where H = o(n1/k).There is a long list of results on sums of five almost equal squares(see[2,3,4,17,24,25,26,27,29,30,35]).In 1996,Liu and Zhan[25]first considered this problem.In 2012,Kumchev and Li[17]obtained the best result at present of this problem:the equation(0.3)has prime solutions satisfying(0.4)with H = nθ/2 for any fixed θ>8/9.They were also the first to obtain results on sums of more than five almost equal squares,where the extra variables are used to reduce the admissible size of H.Let θk,s denote the least exponent θ such that(0.3)and(0.4)with H = nθ/k can be solved for sufficiently large n∈Hk,s whenever θ>θk,s.Kumchev and Li[17]proved that θ2,s ≤19/24 when s ≥17.The lower bound on s in this theorem was reduced to s ≥ 7 in a recent paper by Wei and Wooley[45],in which those authors also established surprisingly strong results for higher values of k:they proved that if s>2k(k-1),one has(?)(?)(?)In 2016,Huang[13]further reduced the bound obtained by Wei and Wooley[45]to θk,s ≤ 19/24 for all k ≥ 3 and s>2k(k-1).In this paper,we use Harman’s sieve methods to break the limitation of the major arcs to θ,and then extend the range of θ.To some extent,we get the best result so far.We also make use of a recent breakthrough by Bourgain,Demeter and Guth[5]to reduce the lower bound on s when k ≥ 4.Our main result further improve Huang’s result and is as follows.Theorem 1 Let k≥2,s≥κ2+κ+1 and 9>31/40.When n∈Hk,s is sufficiently large,equation(0.3)has solutions in primes p1,…,ps satisfying(0.4)with H = nθ/k.The Waring-Goldbach problem with exceptional sets is also an important prob-lem in Number Theory,the reader can refer to the references[17,28,31,38]for the history of this problem.In the same paper,Wei and Wooley[45]also estab-lished results on solubility for" almost all" n which was improved by Huang[13]and on the number of exceptions for representations by six almost equal squares of primes.It is not surprising that by adapting the ideas in[45,§9],our methods lead also to improvements on these two results with the help of Theorem 1.We have the following theorems.Theorem2 Let κ≥2,s>k(k + 1)/2,θ>31/40,and N→∞.There is a fixedδ>0 such that the equation(0.3)has solutions in primes p1,…,ps satisfying(0.4)with H = nθ/k for all but O(N1-δ)integers n ≤ N subject to n∈ Hκ,s(and,whenκ = 3 and s =7,also 9(?)n).Let E6(N;H)denote the number of integer n such that a.|n-N|≤HN1/2,b.n≡ 6(mod 24),c.the equation(0.3)with k = 2 and s= 6 has no solutions in primes P1,...,p6 satisfying(0.4).Theorem 3 Let θ>31/40,and N→∞.There is a fixed δ>0 such that(?)...