伟大的数学思想家—黎曼

   记得，五十多年前，中科院数学所五学科组（几何、数论、拓扑，…，合用一个办公室）的老同学告诉袁萌；他们每天都要“审查”数十封群众来信，其中多半是官关于证明哥德巴赫猜想的稿件，要求给予审查。

当然，毫无例外，来稿都是借助初等数论的证明方法，无一成功。

    时至今日，反动文人王晓明及其团伙，大力宣扬、鼓吹哥德巴赫猜想属于初等数论问题，他们已经解决。

   但是，实际情况是，哥德巴赫问题与黎曼猜想有关。黎曼的数学思想如何伟大，且听下回分解。

   请读者预习本文附件关于黎曼Zeta函数的知识。

袁萌 陈启清  2月16日

附件：

The Riemann Zeta Function

David Jekel

June 6, 2013

In 1859, Bernhard Riemann published an eight-page paper, in which he estimated “the number of prime numbers less than a given magnitude” using a certain meromorphic function on C. But Riemann did not fully explain his proofs; it took decades for mathematicians to verify his results, and to this day we have not proved some of his estimates on the roots of ξ. Even Riemann did not prove that all the zeros of ξ lie on the line Re(z) = 1 2. This conjecture is called the Riemann hypothesis and is considered by many the greatest unsolved problem in mathematics. H. M. Edwards’ book Riemann’s Zeta Function [1] explains the historical context of Riemann’s paper, Riemann’s methods and results, and the subsequent work that has been done to verify and extend Riemann’s theory. The rst chapter gives historical background and explains each section of Riemann’s paper. The rest of the book traces later historical developments and justies Riemann’s statements. This paper will summarize the rst three chapters of Edwards. My paper can serve as an introduction to Riemann’s zeta function, with proofs of some of the main formulae, for advanced undergraduates familiar with the rudiments of complex analysis. I use the term “summarize” loosely; in some sections my discussion will actually include more explanation and justication, while in others I will only give the main points. The paper will focus on Riemann’s denition of ζ, the functional equation, and the relationship between ζ and primes, culminating in a thorough discussion of von Mangoldt’s formula.

1

Contents

1 Preliminaries        3

2 Denition of the Zeta Function   3

2.1 Motivation: The Dirichlet Series . .. . . 4

2.2 Integral Formula . . . . . 4 2.3 Denition of ζ . .  . 5

3 The Functional Equation 6 3.1 First Proof . . .  . 7

3.2 Second Proof . .  . 8

4 ξ and its Product Expansion 9 4.1 The Product Expansion . . . . . 9

4.2 Proof by Hadamard’s Theorem .  . . . . . . 10

5 Zeta and Primes: Euler’s Product Formula 12

6 Riemann’s Main Formula: Summary     13

7 Von Mangoldt’s Formula 15 7.1 First Evaluation of the Integral . .  . . 15

7.2 Second Evaluation of the Integral . .  17

7.3 Termwise Evaluation over ρ . . . . . . . 19

7.4 Von Mangoldt’s and Riemann’s Formulae .. . 22