On Sums Of A Prime And Three Squares Of Primes In Short Intervals

The Waring-Goldbach problem seeks to represent positive integers satisfying necessary congruence conditions by powers of primes, the ternary and binary Goldbach problems are just liner examples of the Waring-Goldbach problem.The circle method of Hardy and Littlewood in combination with the estimates of Vinogiadov for exponential sums over primes gives an affirmative answer to the general Waring-Goldbach problem, and the results before 1965 was summarized in Hua's book "Additive Theory on Prime Numbers". After that, especially in resent years, new ideas in the circle method, sieves, and exponential sums arc incorporated into the Waring-Goldbach problem, and hence give remarkable advances.Waring-Goldbach problems in short intervals have appealed to many authors and have been investigated, among which Goldbach-Vinogradov theorem with almost equal prime variables may be the most famous one .Different from linear case, non-linear cases have to treat, the enlarged major arcs when applying the circle method. In order to avoid this difficulty, Liu and Zhan [11] first studied the quadratic case assuming the Generalized Riemann Hypothesis (GRE). More precisely, they showed that under GRE each large integer N≡5(mod24) can be written aswhere U=N^{1/2-δ+ε},δ=(?). Later Bauer [2] unconditionally showed that the formula (0.1) holds true for U = N^1/2-δ, whereδ≥0 and its exact value depends on the constants in the Deuring-Heilbronn phenomenon, and is not numerically determined.In 1998 Liu and Zhan [11] found the new approach to treat the enlarged major arcs in which the possible existence of Siegel zero does not have special influence, and hence the During-Heibornn phenomenon can be avoided. Due to this approach, they unconditionally obtained that (0.1) is true for U = N^{1/2-1/50+ε}. With the development of this approach, the exponent (?) has subsequently reduced to (?) by Baucr and then to (?) by Guang-Shi L(?).In this paper we study one Waring-Goldbach problem in short intervals .We establish the following result as a short interval version of Hua's theorem on the sum of a prime and three squares of primes (see [5]).Theorem For each sufficiently large even integer not congruent to 0(mod 3), the equation in prime variableshas solution for U = (?),...