Calculating Riemann Zeta function

[Bill Foster]

Homework Statement

Using method of Euler, calculate $$\zeta(4)$$, the Riemann Zeta function of 4th order.

Homework Equations

$$\zeta(s)=\sum_{n=1}^\infty \frac{1}{n^s}$$

Finding $$\zeta(2)$$:

$$\zeta(2)=\sum_{n=1}^\infty \frac{1}{n^s}=\frac{1}{1^2}+\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+\frac{1}{5^2}+\frac{1}{6^2}+...$$

$$\sin{x}=x-\frac{1}{3!}x^3+\frac{1}{5!}x^5+...$$
$$\frac{\sin{x}}{x}=1-\frac{1}{3!}x^2+\frac{1}{5!}x^4+...$$

$$\frac{\sin{x}}{x}=\left(1-\left(\frac{x}{\pi}\right)^2\right)\left(1-\left(\frac{x}{2\pi}\right)^2\right)\left(1-\left(\frac{x}{3\pi}\right)^2\right)\left(1-\left(\frac{x}{4\pi}\right)^2\right)\left(1-\left(\frac{x}{5\pi}\right)^2\right)...$$

From the above, the coefficients of $$x^2$$ are:

$$\frac{1}{\pi^2}+\frac{1}{2^2\pi^2}+\frac{1}{3^2\pi^2}+\frac{1}{4^2\pi^2}+\frac{1}{5^2\pi^2}+...$$

Now equate these coefficients to $$x^2$$ in the sine function series:

$$\frac{1}{\pi^2}+\frac{1}{2^2\pi^2}+\frac{1}{3^2\pi^2}+\frac{1}{4^2\pi^2}+\frac{1}{5^2\pi^2}+...=\frac{1}{3!}$$
$$\frac{1}{1}+\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+\frac{1}{5^2}+...=\frac{\pi^2}{3!}$$
$$\zeta(2)=\frac{\pi^2}{6}$$

The Attempt at a Solution

I get the following for the coefficients of $$x^4$$:

$$\frac{1}{2^2\pi^4}+\frac{1}{3^2\pi^4}+\frac{1}{2^23^2\pi^4}+\frac{1}{4^2\pi^4}+\frac{1}{2^24^2\pi^4}+\frac{1}{3^24^2\pi^4}+...=\frac{1}{\pi^4}\left(\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{2^23^2}+\frac{1}{4^2}+\frac{1}{2^24^2}+\frac{1}{3^24^2}+...\right)$$

The problem is, how do I get $$\zeta(4)=\frac{1}{1^4}+\frac{1}{2^4}+\frac{1}{3^4}+\frac{1}{4^4}+\frac{1}{5^4}+\frac{1}{6^4}+...$$ out of that sum?

 

[mighty2000]

To calculate \zeta(4) using the method of Euler, you can follow these steps:

1. Start with the definition of the Riemann Zeta function: \zeta(s)=\sum_{n=1}^\infty \frac{1}{n^s}

2. Substitute s=4 in the above equation to get: \zeta(4)=\sum_{n=1}^\infty \frac{1}{n^4}

3. Use the fact that \sin{x}=x-\frac{1}{3!}x^3+\frac{1}{5!}x^5+... and \frac{\sin{x}}{x}=1-\frac{1}{3!}x^2+\frac{1}{5!}x^4+... to expand the series for \frac{\sin{x}}{x}.

4. Equate the coefficients of x^4 in the expanded series for \frac{\sin{x}}{x} and \zeta(4).

5. Simplify the equation to get: \zeta(4)=\frac{1}{1^4}+\frac{1}{2^4}+\frac{1}{3^4}+\frac{1}{4^4}+\frac{1}{5^4}+\frac{1}{6^4}+...

6. Use the formula for the sum of a geometric series to simplify the above equation and get the final answer: \zeta(4)=\frac{\pi^4}{90}.

So, the final solution is \zeta(4)=\frac{\pi^4}{90}.