ON ESTIMATIONS OF TRIGONOMETRIC SUMS OVER PRIMES IN SHORT INTERVALS(Ⅱ)

Let α be a real number,x≥A≥2, e（θ） = e, Λ（n） be Mangoldt’s function, and S（α; x, A） = sum from x-A<n≤x (Λ（n）e（nα）.In this paper, the two following results are proved by a purely analytic method. （i） Let ε bean arbitrarily small positive number and x≤A≤x. Then for any given positive c,there exist positive cand csuch that S（α/q +λ; x,A）?A（logx）, provided that （α,q） = 1,1≤q≤logx, and Alogx<| λ |≤(qlogx); （ii） Let N be a sufficiently large odd integer, andU = N. Then the Diophantine equation with prime variables N = p+ p+ pis solv-able for N/3 - U<p≤N/3 + U, j= 1, 2, 3, and there is an asymptotic formula for thenumber of its solutions.