The Riemann zeta function is defined as

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

for $R e(z)>1$, and by analytic continuation to the rest of $\mathbb{C}$ except at singular points. The integral representation of ( $\dagger$ ) for $\operatorname{Re}(z)>1$ is given by

$$\zeta(z)=\frac{1}{\Gamma(z)} \int_{0}^{\infty} \frac{t^{z-1}}{e^{t}-1} d t$$

where $\Gamma$ is the Gamma function.

(a) The Hankel representation is defined as

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

Explain briefly why this representation gives an analytic continuation of $\zeta(z)$ as defined in ( $\ddagger$ ) to all $z$ other than $z=1$, using a diagram to illustrate what is meant by the upper limit of the integral in $(\star)$.

[You may assume $\Gamma(z) \Gamma(1-z)=\pi / \sin (\pi z)$.]

(b) Find

$$\operatorname{Res}\left(\frac{t^{-z}}{e^{-t}-1}, t=2 \pi i n\right)$$

where $n$ is an integer and the poles are simple.

(c) By considering

$$\int_{\gamma} \frac{t^{-z}}{e^{-t}-1} d t$$

where $\gamma$ is a suitably modified Hankel contour and using the result of part (b), derive the reflection formula:

$$\zeta(1-z)=2^{1-z} \pi^{-z} \cos \left(\frac{1}{2} \pi z\right) \Gamma(z) \zeta(z)$$