| The Waring-Goldbach problem concerns how to represent positive integers N that satisfy some necessary congruence conditions by powers of primes. The Gold-bach's Conjecture and the ternary Goldbach problem are well-known examples of linear Waring-Goldbach problem. The circle method of Hardy and Littlewood in combination with the estimates of Vinogradov for exponential sums over primes give a basic tool for studying the Waring-Goldbach problem.Waring-Goldbach problem in short intervals has inspired many mathematicians with researching fervor since 1930's. Many valuable results on this subject, and related topics have been obtained, among which is the most famous Goldbach-Vinogradov's Theorem with almost equal prime variables.Different from the linear case. non-linear Waring-Goldbach problem is much more difficult. since one has to handle the enlarged major arcs to get a better bound on the minor arcs. For this purpose one can not employ the Siegel Walfisz's Theorem directly, so that the integral over the major arcs can not be calculated by classical method. In order to deal with this tough situation, Jianya Liu and Tao Zhan [12] first studied the quadratic case, with the additional assumption of the Generalized Riemann Hypothesis (GRH). More precisely, they showed that under GRH each large integer N≡5 (mod 24) can be written as where U= N1/2δ+ε, and 5=1/20.Later, based on the method of enlarged major arcs from Montgomery and Vaughan [18], Bauer [1] showed that the formula (1) holds for U= N1/2-δwith no additional assumptions, whereδ(> 0) depends on the constant appears in the Deuring-Heilbronn's phenomenon, whose exact value is beyond precise calculation.How to handle the enlarged major arcs in non-linear Waring-Goldbach problem without assuming the GRH or the Deuring-Heilbronn's phenomenon became a major problem. In 1998, Jianya Liu and Tao Zhan [13] found a new approach to handle the enlarged major arcs, in which the possible existence of Siegel zero does not have any particular influence anymore, and hence the Deuring-Heilbronn's phenomenon can be avoided. Through this approach, they showed that (1) is true in general for U=N1/2 1/50+εwith no additional assumptions. In 2003, Jianya Liu [9] introduced the iterative method to treat the major arcs, and this method can be applied to estimate the integral over the major arcs effectively. Until now, the best result on this problem isδ= 1/20, which was obtained by Jianya Liu, Guangshi Lu, and Tao Zhan [11].In this paper, we will prove Theorem 1. For each, sufficiently large integer N≡1 (mod 2), the equation in prime variables is solvable for y = N17/54+εWe prove this theorem by circle method. We first represent the region[1/Q,1+ 1/Q] by unions of the major arcs and the minor arcs. Then we will show that the main term is given by the integral over the major arcs, while the integral over the minor arcs only contributes to the error term.First, we set Define the major arcs M as Minor arcs are the complement of M. in [1/Q,1+1/Q], and we divide them into C(M) and R, where and let R be the complement of M and C(M) in [1/Q,1+1/Q]. So that Therefore, in order to prove Theorem 1, it is sufficient to show that Here. S(a) and T(a) denote the corresponding exponential sums over prime vari-ables, whose definitions can be found in Chapter 1.When handling major arcs M, we apply the iterative method and the following hybrid mean-value estimate in [9] to establish the following asymptotic formula,Proposition 2. Let M be defined as above. Then, for any A＞0, where and the singular series satisfies 1(?)(?)(N)< 1 for N≡1 (mod 2).In estimating integral over the minor arcs, the following new estimate for ex-ponential sums over prines in short intervals of Liu-Lu-Zhan [11] plays an important role. Actually we will establish the followingProposition 3. Let C(M) and R be as above. Then we have...
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