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

Let α be a real number, x≥A≥2, e(θ) = e^(2xiθ), and suppose Λ(n) is Mangoldt's func-tion. In this paper the following result is mainly proved: Let ε be an arbitrarily small po-sitive number, and x^(91/(96+ε))≤A≤x. Then for any given positive c, there exists a positive c_1such that for A^(-1)log^cx ≤|α| ≤(logx)^(-c_1) there exitss sum from x-A<n≤x (Λ(n)e(nα)? A(logx)^(-c).

Let α be a real number,x≥A≥2, e（θ） = e2πiθ, 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 x91/96+ε≤A≤x. Then for any given positive c,there exist positive c1 and c2 such that S（α/q +λ; x,A）?A（logx）-c, provided that （α,q） = 1,1≤q≤logc1x, and A-1logc2x<| λ |≤(qlogc1x)-1; （ii） Let N be a sufficiently large odd integer, andU = N91/96+ε. Then the Diophantine equation with prime variables N = p1 + p2 + p3 is solv-able for N/3 - U<pj≤N/3 + U, j= 1, 2, 3, and there is an asymptotic formula for thenumber of its solutions.