ON ESTIMATIONS OF TRIGONOMETRIC SUMS OVER PRIMES IN SHORT INTERVALS(I)
ON ESTIMATIONS OF TRIGONOMETRIC SUMS OVER PRIMES IN SHORT INTERVALS(I)
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DOI：
10.1360/ya1989-32-4-408
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发表时间：
1989-04
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通讯作者：
C. Pan
中科院分区：
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作者：
C. Pan
设α为真实的数，x≥A≥2，e（θ）= e2 πiθ，Λ（n）为Mangoldt函数，S（α; x，A）= sum from x-An≤x（Λ（n）e（nα）.本文用纯解析方法证明了以下两个结果. (i)设e为任意小正数，x~（91/96+e）≤A≤x.则对任意给定的正c，存在正c_1和c_2使得S（α/q +λ; x，A）？A（logx）~（-c），条件是（α，q）= 1，1 ≤q≤log~（c_1）x，且A~（-1）log~（c_2）x| λ |≤（qlog~（c_1）x）~（-1）;（ii）设N为充分大的奇整数，U = N~（91/96+e）.则素变元丢番图方程N = p_1 + p_2 + p_3对N/3 - Up_j≤N/3 + U，j= 1，2，3是可解的，并有其解的个数的渐近公式。
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.