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Let α be a real number,x≥A≥2, e（θ） = e<sup>2πiθ</sup>, 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