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
About this Honors Thesis
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