唧唧堂编辑二朵2014-01-09 3:57 AM

Major Arcs For Goldbach's Problem 哥德巴赫定理的主要弧

Abstract The ternary Goldbach conjecture states that every odd number n≥7 is the sum of three primes. The estimation of the Fourier series ∑p≤xe(αp) and related sums has been central to the study of the problem since Hardy and Littlewood (1923). Here we show how to estimate such Fourier series for α in the so-called major arcs, i.e., for α close to a rational of small denominator. This is part of the author's proof of the ternary Goldbach conjecture. In contrast to most previous work on the subject, we will rely on a finite verification of the Generalized Riemann Hypothesis up to a bounded conductor and bounded height, rather than on zero-free regions. We apply a rigorous verification due to D. Platt; the results we obtain are both rigorous and unconditional. The main point of the paper will be the development of estimates on parabolic cylinder functions that make it possible to use smoothing functions based on the Gaussian. The generality of our explicit formulas will allow us to work with a wide variety of such functions. 论文摘要 三元哥德巴赫猜想指出,每个奇数都(n ≥ 7)是三个素数之和。继Hardy和Littlewood之后( 1923),对傅立叶级数Σp ≤xe(αp)与其和的估计对该问题的研究来说极为重要。我们将在这里展示如何估算有关α在主要弧内的傅里叶级数,比如,当α逼近小分母有理数时。这是作者对三元哥德巴赫猜想的部分证明。与大部分以前的工作相比,在这个问题上,我们将凭借一个有界导体和有界高度而非无零点区域的广义黎曼假设的有限验证。我们依据D. Platt采用了严格验证;且获得的结果都是严谨的和无条件限制的。本文的重点将是对抛物柱面函数估计的完善,这使人们有可能基于高斯函数来使用平滑函数。该显式表达式的一般性原则将使我们能够在研究中借助各种各样这一类函数。

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