The Riemann zeta function is defined by
ζR(s)=n=1∑∞n−s
for Re(s)>1.
Show that
ζR(s)=Γ(s)1∫0∞et−1ts−1dt
Let I(s) be defined by
I(s)=2πiΓ(1−s)∫Ce−t−1ts−1dt
where C is the Hankel contour.
Show that I(s) provides an analytic continuation of ζR(s) for a range of s which should be determined.
Hence evaluate ζR(−1).