The Riemann zeta function is defined by

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

for $\operatorname{Re}(s)>1$.

Show that

$$\zeta_{R}(s)=\frac{1}{\Gamma(s)} \int_{0}^{\infty} \frac{t^{s-1}}{e^{t}-1} d t$$

Let $I(s)$ be defined by

$$I(s)=\frac{\Gamma(1-s)}{2 \pi i} \int_{C} \frac{t^{s-1}}{e^{-t}-1} d t$$

where $C$ is the Hankel contour.

Show that $I(s)$ provides an analytic continuation of $\zeta_{R}(s)$ for a range of $s$ which should be determined.

Hence evaluate $\zeta_{R}(-1)$.