Riemann Zeta Function and pi..

Question

Why Does PI keep showing up in the Zeta function ?

I am a newbie to this topic (just saw a video on youtube)... I am thus tempted to know more about it.

Answer 1

First off, only the even values of the Zeta function have known closed form, so we can't say that $\pi$ will appear for an arbitrary value.

I think one of the most compelling arguments for the appearance of $\pi$ is Euler's solution to the Basel problem, which is finding the value of $\zeta (2) $

$$\frac{\sin(x)}{x} = \sum_{n=0}^\infty \frac{(-1)^n x^{2n}}{(2n+1)!} $$ This much is known by Taylor expanding $\sin(x)$ and then dividing by $x$. Similar to a polynomial, we can write this function as a product of its zero's (in similar fashion to the Fundamental Theorem of Algebra, but in an infinite case. Euler was unjustified here, but Weierstrass's Theorem later validated the approach)

$$\frac{\sin(x)}{x} = (1-\frac{x}{\pi})(1+\frac{x}{\pi})(1-\frac{x}{2\pi})(1+\frac{x}{2\pi})... $$

$$1 - \frac{x^2}{3!} + \frac{x^4}{5!}+... = (1-\frac{x^2}{\pi^2})(1-\frac {x^2}{2^2 \pi^2})(1-\frac{x^2}{3^2 \pi^2})... $$ Expanding this: $$1 - \frac{x^2}{3!} + O(x^4) = 1 - \frac{x^2}{\pi^2} \sum_{n=1}^\infty \frac1 {n^2} + O(x^4) $$

Equating $x^2$ terms on each side:

$$\sum_{n=1}^\infty \frac1 {n^2} = \pi^2/6$$

This method can be extended well to $\zeta(2^n) $

Answer 2

This is very intriguing at first, especially if you are not familiar with Taylor series. Turns out Euler's approach to justify this appearance is tied to the fact that we can represent a trigonometric function as such:

$$\sin(x) = x -\frac{x^3}{3!}+\frac{x^5}{5!}-\frac{x^7}{7!}+...$$

We can then do some manipulation to this (which you can read further here, it's called the Basel problem) and because of the fact that $\sin(x)$ takes in as input radians, which are in terms of $\pi$, it's natural that you will get $\pi$ in a result of an infinite sum like with the Reimann Zeta function. Of course I'm skipping over many steps, but if you're looking for an overview, the Taylor series really ties together this idea of an infinite sum of polynomials and a trig function.