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

Abstract

Let α be a real number,x≥A≥2, e(θ) = e~(2πiθ), Λ(n) be Mangoldt's function, andS(α; x, A) = sum from x-An≤x (Λ(n)e(nα).In this paper, the two following results are proved by a purely analytic method. (i) Let e bean arbitrarily small positive number and x~(91/96+e)≤A≤x. Then for any given positive c,there exist positive c_1 and c_2 such that S(α/q +λ; x,A)?A(logx)~(-c), provided that (α,q) = 1,1≤q≤log~(c_1)x, and A~(-1)log~(c_2)x| λ |≤(qlog~(c_1)x)~(-1); (ii) Let N be a sufficiently large odd integer, andU = N~(91/96+e). Then the Diophantine equation with prime variables N = p_1 + p_2 + p_3 is solv-able for N/3 - Up_j≤N/3 + U, j= 1, 2, 3, and there is an asymptotic formula for thenumber of its solutions.