The Waring-Goldbach Problems In Short Intervals

Waring-Goldbach problems in short intervals have appealed to many authors and have been investigated, among which the Goldbach-Vinogradov theorem with almost equal prime variables may be the most famous one (see for example [1], [2], [3] and[4]).Different from this 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 [15] first studied the quadratic case assuming the Generalized Riemann Hypothesis (GRH). More precisely, they showed that under GRH each large integer TV ≡ 5(mod 24) can be written asN=p₁²+p₂² + p₃²+p₄²+p₅² (0.1)whereLater Bauer [16] unconditionally showed that the formula (0.1) holds true for U =N^(1/2)-δ, where δ ≥0 and its exact value depends on the constants in the Deuring-Heilbronn phenomenon, and is not numerically determined.In 1998 Zhan and Liu [15] 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 Dcuring-Heilbronn phenomenon can be avoided. Due to this approach, they obtained that (0.1) is true for U = N⁽(1/2)-(1/(50)+ε). With the development of this approach, the exponent (1/2) —(1/50) has subsequently reduced to (1/2) — (19/850) by Bauer [9] and then to (1/2) - (1/35) by the first author of [17].In this paper we study two Waring-Goldbach problems in short intervals.