Chapter 1.15, Problem 22P

The series ${\displaystyle {\sum }_{n=1}^{\infty }1/n^8,s>1}$, is called the Riemann Zeta function, $\zeta (s)$.

(In Problem 14.2(a) you found $\zeta (2)={\pi }^2/6$.

When n is an even integer, these series can be summed exactly in terms of $\pi$.) By computer or tables, find

(a) $\zeta (4)={\displaystyle \underset{n=1}{\overset{\infty }{\sum }}\frac{1}{n^4}}$

(b) $\zeta (3)={\displaystyle \underset{n=1}{\overset{\infty }{\sum }}\frac{1}{n^3}}$

(c) $\zeta (\frac{3}{2})={\displaystyle \underset{n=1}{\overset{\infty }{\sum }}\frac{1}{n^{3/2}}}$