# Riemann Zeta Function of 4

## Theorem

The Riemann zeta function of $4$ is given by:

 $\displaystyle \map \zeta 4$ $=$ $\displaystyle \dfrac 1 {1^4} + \dfrac 1 {2^4} + \dfrac 1 {3^4} + \dfrac 1 {4^4} + \cdots$ $\displaystyle$ $=$ $\displaystyle \dfrac {\pi^4} {90}$ $\displaystyle$ $\approx$ $\displaystyle 1 \cdotp 08232 \, 3 \ldots$

## Proof 1

By Fourier Series of Fourth Power of x, for $x \in \closedint {-\pi} \pi$:

$\displaystyle x^4 = \frac {\pi^4} 5 + \sum_{n \mathop = 1}^\infty \frac {8 n^2 \pi^2 - 48} {n^4} \, \map \cos {n \pi} \, \map \cos {n x}$

Setting $x = \pi$:

 $\displaystyle \pi^4$ $=$ $\displaystyle \frac {\pi^4} 5 + \sum_{n \mathop = 1}^\infty \frac {8 n^2 \pi^2 - 48} {n^4} \, \map {\cos^2} {n \pi}$ $\displaystyle \leadsto \ \$ $\displaystyle \frac {4 \pi^4} 5$ $=$ $\displaystyle \sum_{n \mathop = 1}^\infty \frac {8 n^2 \pi^2} {n^4} - \sum_{n \mathop = 1}^\infty \frac {48} {n^4}$ Cosine of Multiple of Pi $\displaystyle \leadsto \ \$ $\displaystyle \frac {\pi^4} 5$ $=$ $\displaystyle 2 \pi^2 \sum_{n \mathop = 1}^\infty \frac 1 {n^2} - 12 \sum_{n \mathop = 1}^\infty \frac 1 {n^4}$ $\displaystyle$ $=$ $\displaystyle \frac {\pi^4} 3 - 12 \sum_{n \mathop = 1}^\infty \frac 1 {n^4}$ Basel Problem $\displaystyle \leadsto \ \$ $\displaystyle 12 \sum_{n \mathop = 1}^\infty \frac 1 {n^4}$ $=$ $\displaystyle \frac {\pi^4} 3 - \frac {\pi^4} 5$ rearranging $\displaystyle$ $=$ $\displaystyle \frac {2 \pi^4} {15}$ $\displaystyle \leadsto \ \$ $\displaystyle \sum_{n \mathop = 1}^\infty \frac 1 {n^4}$ $=$ $\displaystyle \frac {\pi^4} {90}$

$\blacksquare$

## Proof 2

By Fourier Series of x squared, for $x \in \left[{- \pi \,.\,.\, \pi}\right]$:

$\displaystyle x^2 = \frac {\pi^2} 3 + \sum_{n \mathop = 1}^\infty \left({\left({-1}\right)^n \frac 4 {n^2} \cos n x}\right)$

Hence:

 $\displaystyle \frac 1 \pi \int_{-\pi}^\pi x^4 \, \mathrm d x$ $=$ $\displaystyle \frac 1 2 \left({\frac {2 \pi^2} 3}\right)^2 + \sum_{n \mathop = 1}^\infty \left({\frac {4 \left({-1}\right)^n} {n^2} }\right)^2$ Parseval's Theorem $\displaystyle \leadsto \ \$ $\displaystyle \frac 2 \pi \int_0^\pi x^4 \, \mathrm d x$ $=$ $\displaystyle \frac {2 \pi^4} 9 + \sum_{n \mathop = 1}^\infty \frac {16} {n^4}$ Definite Integral of Even Function $\displaystyle \leadsto \ \$ $\displaystyle \frac {2 \pi^4} 5$ $=$ $\displaystyle \frac {2 \pi^4} 9 + \sum_{n \mathop = 1}^\infty \frac {16} {n^4}$ $\displaystyle \leadsto \ \$ $\displaystyle \frac {8 \pi^4} {45}$ $=$ $\displaystyle \sum_{n \mathop = 1}^\infty \frac {16} {n^4}$ $\displaystyle \leadsto \ \$ $\displaystyle \sum_{n \mathop = 1}^\infty \frac 1 {n^4}$ $=$ $\displaystyle \frac {\pi^4} {90}$

$\blacksquare$

## Proof 4

 $\displaystyle \map \zeta 4$ $=$ $\displaystyle \paren {-1}^3 \dfrac {B_4 2^3 \pi^4} {4!}$ Riemann Zeta Function at Even Integers $\displaystyle$ $=$ $\displaystyle \paren {-1}^3 \paren {-\dfrac 1 {30} } \dfrac {2^3 \pi^4} {4!}$ Definition of Sequence of Bernoulli Numbers $\displaystyle$ $=$ $\displaystyle \paren {\dfrac 1 {30} } \paren {\dfrac 8 {24} } \pi^4$ Definition of Factorial $\displaystyle$ $=$ $\displaystyle \dfrac {\pi^4} {90}$ simplifying

$\blacksquare$

The decimal expansion can be found by an application of arithmetic.

## Historical Note

The Riemann Zeta Function of 4 was solved by Leonhard Euler, using the same technique as for the Basel Problem.

If only my brother were alive now.
-- Johann Bernoulli