数论中的组合恒等式及其应用.doc

PAGE I PAGE I 数论中的组合恒等式及其应用 摘 要 数论是 纯粹数学的分支之一，主要研究 整数的性质。按研究方法来看，数论大致可分为 初等数论、 代数数论、 解析数论、 概率数论等等。解析数论的方要有 复变 积分法、圆法、 筛法、指数和方法等等。组合恒等式是筛法的一个重要分支，它起源于上世纪30年代Vinogradov关于线性素变数指数和的研究，接着被Linnik成功地运用，后来又被许多数学家发展，例如在上世纪70年代和80年代，Vaughan和Heath-Brown先后提出了以他们名字命名的恒等式。如今，这些恒等式在某些与素数相关的问题的研究中起着极其关键的作用。 本文首先介绍Riemann对于解析数论发展的重大贡献及Vaughan恒等式的理论背景，接下来给出Vaughan恒等式的解析方法证明及初等方法证明，之后作为Vaughan恒等式的推广，我们对Heath-Brown恒等式进行推导，最后简要介绍Vaughan恒等式的一个应用。本文主要研究方法为解析方法，基本内容属于解析数论范畴。 关 键 词：Vaughan恒等式;Heath-Brown恒等式；解析数论 西安交通大学本科毕业论文 PAGE VI PAGE VI PAGE I PAGE I ABSTRACT The number theory is one of the branches of pure mathematics, mainly studying the nature of the integer. According to the research method, the number theory can be divided into elementary number theory，algebraic number theory, analytic number theory, computational number theory and so on.Among them, the method of analytic numbering mainly includes complex integral method, circle method, sieve method, index and method.The combinatorial identity is an important branch of the sieve method, which originated from Vinogradov’s research on the exponential sums over primes in 1930s, and then successfully used by Linnik, and later by many mathematicians, such as in 1970s and 1980s, Vaughan and Heath-Brown have proposed identifiers named after them. Nowadays, these identities play a crucial role in the study of some prime-related problems. This paper first introduces the significant contribution of Riemann to the development of analytic number theory and the theoretical background of Vaughans identity. Then we give the proof of the analytical method of Vaughans identity and the proof of the elementary method. Then we generalize the Heath-Brown identity as a generalization of the Vaughan identity. Finally, we briefly introduce an application of the Vaughan identity.The main research method is analytic method, the basic content belongs to the analytic number theory category. KEY WORDS: Vaughan identity; H