An Exposition of the Functional Equation for the Riemann Zeta Function and its Values at Integers Open Access

Guo, Jiaqi (2013)

Permanent URL: https://etd.library.emory.edu/concern/etds/d504rm07r?locale=en
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Abstract

This paper is an exposition of the development of the zeta function, as well as proving and deriving the essential elements which lead to the functional equation of the Riemann zeta function. We start from the historical background and motivation of defining the zeta function. As Euler first defined this function for the real numbers, he utilized it to prove that there exist infinitely many primes. In addition, a proof for ζ(2), the solution to Basel Problem, was also included in this paper. Then we move on to the Riemann zeta function and its analytic continuation on the whole complex plane. Finally, with an objective to evaluate the zeta function for all the positive even integers, we examine the Bernoulli numbers and their connection with the zeta function.

Table of Contents

Contents 1 Introduction and Statement of the Formula 1 1.1 Preliminary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2 Historical Background 2 2.1 Zeta of the Real . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2.2 Riemann Zeta Function . . . . . . . . . . . . . . . . . . . . . . . . . . 2 3 Applications of the Zeta Function by Euler 3 3.1 Proof of the Innitude of Prime Numbers . . . . . . . . . . . . . . . . 3 3.2 Basel Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 4 Riemann Zeta Function 7 4.1 Gamma Function and its Properties . . . . . . . . . . . . . . . . . . . 7 4.2 From Gamma to Zeta Function . . . . . . . . . . . . . . . . . . . . . 9 4.3 Poisson's Summation Formula . . . . . . . . . . . . . . . . . . . . . . 10 4.4 Transformation Law for Theta Function . . . . . . . . . . . . . . . . 13 4.5 Functional Equation of Riemann Zeta Function . . . . . . . . . . . . 16 5 Riemann Zeta Function and Bernoulli Numbers 17 5.1 Values of the Riemann Zeta Function . . . . . . . . . . . . . . . . . . 17 5.2 Bernoulli Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 5.3 Bernoulli Numbers and the Zeta Function . . . . . . . . . . . . . . . 20 6 Conclusion 22 7 Appendix 23

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