= Solution
For $\Re s>1$, the <Riemann zeta function> is the <absolutely convergent> <Dirichlet series> $\zeta(s)=\sum_{n\ge1}n^{-s}$. Apply <Abel summation> to the tail, with counting function $\lfloor w\rfloor-\lfloor x\rfloor$. The upper boundary term tends to zero, giving
$$
\sum_{n>x}n^{-s}=-\lfloor x\rfloor x^{-s}+s\int_x^\infty\lfloor w\rfloor w^{-s-1}\,dw.
$$
Substitute $\lfloor w\rfloor=w-\{w\}$ and integrate the $w$ term. This proves the <fractional-part continuation formula for the Riemann zeta function>
$$
\boxed{\zeta(s)=\sum_{n\le x}n^{-s}+\frac{x^{1-s}}{s-1}+\{x\}x^{-s}-s\int_x^\infty\{w\}w^{-s-1}\,dw.}
$$
The 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 \b[exactly one <pole> in $\Re s>0$: a simple <pole> at $s=1$ of <residue> one.]
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