陈景润其人与《陈景润文集》

陈景润其人与《陈景润文集》潘承彪在我最不愿意回忆往事的时候，江西教育出版社于景润离开我们两周年的前夕寄来了我所盼望的他的文集。书中印有我十分熟悉的《关于哥德巴赫问题和筛法》一文手稿中一页的封面，设计古朴、典雅、凝重。我凝视良久，思绪万千，实在不愿去翻...

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

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).

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

In this paper the following result is proved: There is an absolute positive integer such that for every large odd integer N the Diophantine equation with prime variables N=p1+p2+p3, N/3-U【pj≤N/3+U, j=1, 2, 3, is solvable for U=N2/3 logc N. Moreover, an asymptotic formula for the number of the solutions is given.

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).

小区间上的素变数三角和估计Ⅱ

设α是实数,x≥A≥2,e(θ)=e^(2xiθ),Λ(n)是Mangoldt函数,以及本文用纯分析方法证明了:(ⅰ)设ε是任给的小正数,x^((91)/(96)+ε)≤A≤x。那么对任给的正数c,一定存在正数c_1,c_2,使当α=a/q+λ满足:(α,q)=1,1≤q≤log^(c_1)x,时,(见定理2);(ⅱ)设N是大奇数,U=N^((91)/(96)+ε),则素变数不定方程N=p_1+p_2+p_3,N/3—U<p_j≤N/3+U,j=1,2,3,一定有解,且解数有渐近公式(见定理3)。