= Solution
For $\operatorname{Re}s>1$, expand $(e^t-1)^{-1}=\sum_{n\ge1}e^{-nt}$. Absolute integrability, bounded near zero by a constant times $t^{\operatorname{Re}s-2}$, permits termwise integration. The <Gamma function> <integral> then gives
$$
\int_0^\infty\frac{t^{s-1}}{e^t-1}\,dt
=\sum_{n\ge1}n^{-s}\Gamma(s),
\qquad \boxed{\zeta_R(s)=\frac1{\Gamma(s)}\int_0^\infty\frac{t^{s-1}}{e^t-1}\,dt}.
$$
Specify the <Hankel contour> to run from the negative real axis below the cut towards zero, circle zero counterclockwise with fixed radius $0<r<2\pi$, and return above the cut. Take $-\pi<\arg t<\pi$. The circle excludes every nonzero pole $2\pi i\mathbb Z$ of the denominator. For $\operatorname{Re}s>1$ its radius can shrink to zero, and the lower and upper rays give respectively $-e^{-i\pi s}\Gamma(s)\zeta_R(s)$ and $e^{i\pi s}\Gamma(s)\zeta_R(s)$. By the <Gamma reflection formula>,
$$
I(s)=\frac{\Gamma(1-s)\Gamma(s)\sin\pi s}{\pi}\zeta_R(s)=\zeta_R(s)
$$
away from integer $s$ in that half-plane, with limiting equality at the removable points.
With the circle radius kept fixed, the contour <integral> is an <entire function> of $s$: the rays decay exponentially and converge locally uniformly, and the circular part stays away from zero. Multiplication by $\Gamma(1-s)$ is analytic for $\operatorname{Re}s<1$ and meromorphic globally. The apparent singularities at positive integers $s\ge2$ are removable because the <integral> vanishes there and it already agrees with the original analytic zeta <function> nearby. Only $s=1$ remains a simple pole. Thus this constructs the <meromorphic continuation> to the whole plane and an <analytic continuation> on $\mathbb C\setminus\{1\}$. Keeping the circular segment is essential when the <integral> is not locally integrable at zero.
At $s=-1$ the two rays cancel, since $t^{-2}$ is single-valued. The Laurent expansion
$$
\frac1{e^{-t}-1}=-\frac1t-\frac12-\frac t{12}+O(t^3)
$$
shows that the residue of $t^{-2}/(e^{-t}-1)$ is $-1/12$. Since $\Gamma(2)=1$, the <residue theorem> gives
$$
\boxed{\zeta_R(-1)=-\frac1{12}}.
$$
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