The Cubic Waring-Goldbach Problem In Short Intervals

In 1742, Goldbach wrote Euler raising the Goldbach Congecture which can be stated as folows:(A) Every odd number which is equal to or greater than 9 is the sum of three odd primes.(B) Every even number which is equal to or greater than 6 is the sum of two odd primes.The odd Goldbach Congecture (A) is also known as three prime theorem which is almost solved by Vinogradov in 1937. He proved that every sufficiently large odd integer is the sum of three primes. The even Goldbach Congecture is still open now.The non-linear Waring problem is also known as Waring-Goldbach problem which concerns the representation of positive integers n by powers of j primes, i.e. where k is some given positive integer. and j = j(k) depends on k. For some fixed k, j is hoped to be as small as possible. It is conjectured that for k> 1 and j> k+1. the above equal ion is solvable. Of course this is a very difficult problem that can not be reach at present. But it is possible to approximate this problem in some way. In this paper, we are interested in the case of k= 3.For the cubic Waring-Goldbach problem, corresponding to the even Goldbach Congecture. the conjecture is that all sufficiently large integers n satisfying some nec-essary congruence conditions, are the sum of four cubes of primes, i.e. This conjecture is more difficult than the even Goldbach Congecture, which is out of reach at. present. There are not many essential advances about this conjecture, but there are indeed approximations to support this conjecture, such as Davenport's result. The theorem of Davenport in [3] asserted that almost all positive integers are the sum of four positive cubes, where "almost all" means that the set of integersε(z) satisfying some necessary conditions and less than z which can not be written as the sum of four cubes of positive integers satisfiesε(z)= o(z).Let (?) be the set of integers n which can be written as (2). In 1949, Roth [26] showed that This result can be viewed as a direct approximation to the above conjecture. In [23], Ren proved that whereβ> 0 is an absolute constant. It therefore follows that the conjecture is true for a positive proportion of positive integers. It is also interesting to know an acceptable numerical value ofβIn another paper of Ren [24], an acceptable value ofβ= 1/320 is obtained.In chapter 1, we obtained the following result in short intervals:Theorem 1.1. Let N be a large integer and (?) as above. Then there exists an absolute constant (?)> 0 such thatThe above result states that for any given sufficiently large N and given Y in the theorem, the conjecture is also true for a positive proportion of positive integers in short interval [N,. N+Y]. We will combine the circle method and the upper bound sieves to prove Theorem 1.1. If j = 5,6,7,8 and k = 3, denote by(z) the set of integers n∈Aj∩[z/2, z] such that n can not be written as (1), where Aj is some congruence conditions defined in (2.2). Hua's theorem in [5]-[6] proved that almost all positive integers n satisfying some necessary conditions are the sum of five cubes of primes. Hua proved thatε5(z) < < z(logz)-A, where A> 0 is an arbitary constant. Some mathematician have studied this probelm since then. The most recent result in this problem is proved by Kumchev [2] in 2005. his result states as follows:If 5≤j≤8, we have where respectively.In the case of j= 5,6,7,8. we investigate this problem with pi taking values in short intervals: whereδj> 0 is a constant, which is hoped to be as large as possible. In chapter 2, our result gives deep insights into Kumchev's results, which is stated as follows.Theorem 1.2. Let j= 5.6,7,8. Aj is defined in (2.2). For any fixedε> 0, the equation (5) withδj= 1/45,1/30,1/25,2/45 is solvable for all integers N∈Aj∩[z/2,2], respectively, but for at most O(z1-ε) exceptions.This result indicates that not only almost all integers satisfying some necessary congruence conditions are the sum of j (j= 5,6,7.8) cubes of primes, but also the primes can take values in short intervals.In the case of j = 9, Hua [5] stated that each sufficiently large odd integer can be written as the sum of nine cubes of primes. Lii and Xu [13] proved that all integers can be written as in (5) withδ9= 1/198, which is as strong as the result under the Generalized Riemann Hypothesis. In the proof of Theorem 1.2, we use the method in [13].