The Riemann zeta function is defined by the sum

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

which converges for $\operatorname{Re} s>1$. Show that

$$\zeta(s)=\frac{1}{\Gamma(s)} \int_{0}^{\infty} \frac{t^{s-1}}{e^{t}-1} d t, \quad \operatorname{Re} s>1$$

The analytic continuation of $\zeta(s)$ is given by the Hankel contour integral

$$\zeta(s)=\frac{\Gamma(1-s)}{2 \pi i} \int_{-\infty}^{0+} \frac{t^{s-1}}{e^{-t}-1} d t$$

Verify that this agrees with the integral $(*)$ above when Re $s>1$ and $s$ is not an integer. [You may assume $\Gamma(s) \Gamma(1-s)=\pi / \sin \pi s$.] What happens when $s=2,3,4, \ldots$ ?

Evaluate $\zeta(0)$. Show that $\left(e^{-t}-1\right)^{-1}+\frac{1}{2}$ is an odd function of $t$ and hence, or otherwise, show that $\zeta(-2 n)=0$ for any positive integer $n$.