= Solution
For $\Re s>1$, the absolutely convergent <Dirichlet series>
$$
\zeta(s)=\sum_{n=1}^\infty n^{-s}
$$
defines the <Riemann zeta function>. To continue it, use <partial summation> in Stieltjes form:
$$
\zeta(s)
=s\int_1^\infty\lfloor x\rfloor x^{-s-1}\,dx
=\frac{s}{s-1}-s\int_1^\infty\{x\}x^{-s-1}\,dx.
$$
Since $0\leq\{x\}<1$, the final integral converges locally uniformly for $\Re s>0$ and is <holomorphic> there. The displayed expression is consequently a <meromorphic function> on that half-plane, with its only pole at $s=1$. Since $s/(s-1)$ has residue one there, so does $\zeta$. Agreement in $\Re s>1$ makes this continuation unique by the <identity theorem>.
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