Past exam of the mathematics course of the University of Cambridge / 2014 / iii / Paper 25 / 1 / a / Solution

For $\Re s>1$, the Riemann zeta function is the absolutely convergent Dirichlet series $\zeta (s)={\sum }_{n\ge 1}{n}^{-s}$. Apply Abel summation to the tail, with counting function $\lfloor w\rfloor -\lfloor x\rfloor$. The upper boundary term tends to zero, givingSubstitute $\lfloor w\rfloor =w-\{w\}$ and integrate the $w$ term. This proves the fractional-part continuation formula for the Riemann zeta functionThe endpoint convention is valid whether or not $x$ is an integer. Since $0\le \{w\}<1$, the last integral converges locally uniformly for $\Re s>0$, including after differentiation on compact subsets. It therefore defines a holomorphic function there. The other terms are entire except for $x^{1-s}/(s-1)$. By the identity theorem for holomorphic functions, the formula supplies a meromorphic continuation with exactly one pole in $\Re s>0$: a simple pole at $s=1$ of residue one.