Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 150 / 3 / a / Solution

For $\sigma >1$, summation over the intervals $[n,n+1)$ givesSince $\lfloor u\rfloor =u-\{u\}$,The integral converges absolutely and defines a holomorphic function for $\sigma >0$. This formula therefore gives the Meromorphic continuation of the Riemann zeta function to the right half-plane, with only a simple pole at $s=1$ and residue one.