Chapter 10.4, Problem 67E

The zeta function The Riemann zeta function is the subject of extensive research and is associated with several renowned unsolved problems. It is defined by $\zeta (x)={\displaystyle \underset{k=1}{\overset{\ne }{\sum }}\frac{1}{k^x}}$. When x is a real number, the zeta function becomes a p series. For even positive integers p, the value of ζ(p) is known exactly. For example,

${\displaystyle \underset{k=1}{\overset{\ne }{\sum }}\frac{1}{k^2}=\frac{{\pi }^2}{6},}{\displaystyle \underset{k=1}{\overset{\ne }{\sum }}\frac{1}{k^4}=\frac{{\pi }^4}{90},\text{and}{\displaystyle \underset{k=1}{\overset{\ne }{\sum }}\frac{1}{k^6}=\frac{{\pi }^6}{945},\dots .}}$

Use the estimation techniques described in the text to approximate ζ(3) and ζ(5) (whose values are not known exactly) with a remainder less than 10⁻³.