$\zeta(0)$ and the cotangent function In preparing some practice problems for my complex analysis students, I stumbled across the following. It is not hard to show, using Liouville's theorem, that
$$\pi\cot(\pi z)=\frac{1}{z}+\sum_{n=1}^\infty\left(\frac{1}{z+n}+\frac{1}{z-n}\right),$$
which implies that
$$-\frac{\pi z}{2}\cot(\pi z)=-\frac{1}{2}+\sum_{k=1}^\infty\zeta(2k)z^{2k},\qquad 0<|z|<1.$$
This formula predicts correctly that $\zeta(0)=-\frac{1}{2}$, and allows to calculate $\zeta(2k)$ as a rational multiple of $\pi^{2k}$ as well (in terms of Bernoulli numbers).
Is there some simple explanation why the above prediction $\zeta(0)=-\frac{1}{2}$ is valid? Perhaps there is a not so simple but still transparent explanation via Eisenstein series.
Added. Just to clarify what I mean by "simple explanation". The second identity above follows directly from the first identity, i.e. from basic principles of complex analysis:
$$-\frac{\pi z}{2}\cot(\pi z)=-\frac{1}{2}+\sum_{n=1}^\infty\frac{z^2}{n^2-z^2}=-\frac{1}{2}+\sum_{n=1}^\infty\sum_{k=1}^\infty\left(\frac{z^2}{n^2}\right)^k
=-\frac{1}{2}+\sum_{k=1}^\infty\zeta(2k)z^{2k}.$$
I would like to see a similar argument, perhaps somewhat more elaborate, that explains why the constant term here happens to be $\zeta(0)$, which seems natural in the light of the other terms.

Here is an explanation based on the Euler-Maclaurin summation formula.
(Or rather, since we'll only ever need two terms of the Euler-Maclaurin summation, it's really more or less just "the trapezoid rule".)
I think it's a good explanation because it sticks to the general structure of the argument outlined in the question, and its "18th-century-friendly" spirit.
First, let's review how to apply Euler-Maclaurin to $\zeta$.
We have
$$
    \begin{align}
        \zeta(s) &=
      \sum_{n=1}^N \frac{1}{n^s}
   + \sum_{n=N+1}^{\infty} \frac{1}{n^s} \\
        &=
      \sum_{n=1}^N \frac{1}{n^s}
   + \int_N^{\infty} \frac{1}{x^s} \, dx
   - \frac12 \frac{1}{N^s}
   + \mathrm{Error} \\
        &=
      \sum_{n=1}^N \frac{1}{n^s}
   + \frac{1}{s-1} \frac{1}{N^{s-1}}
   - \frac12 \frac{1}{N^s}
   + \mathrm{Error}
   \tag{1}. \\
    \end{align}
$$
We can say more about the error term later - for now, all we need to know is that $\mathrm{Error} = O ( \frac{1}{N^{\operatorname{Re}(s)+1}} )$ as $N \to \infty$, for each $s$.
The key thing to note is that, even though the computation initially required $\operatorname{Re}(s)>1$ in order to be valid, in fact the "equation"
$$
    \zeta(s) =
     \sum_{n=1}^N \frac{1}{n^s}
  + \frac{1}{s-1} \frac{1}{N^{s-1}}
  - \frac12 \frac{1}{N^s}
  + O \left( \frac{1}{N^{\operatorname{Re}(s)+1}} \right)
  \tag{2}
$$
has a unique constant solution $\zeta(s)$, for each $s$ with $\operatorname{Re}(s) > -1$ and $s \ne 1$.
This is one way to define $\zeta(s)$ for all such $s$.
In particular, we can plug in $0$ and immediately find that $\zeta(0) = -\frac12$.
What about the function $f(z) = -\frac{\pi z}{2} \cot(\pi z)$? We can also apply Euler-Maclaurin to $f$ in the same way:
$$
    \begin{align}
        f(z) &=
      -\frac12
   + \sum_{n=1}^N \frac{z^2}{n^2-z^2}
   + \sum_{n=N+1}^{\infty} \frac{z^2}{n^2-z^2} \\
        &=
      -\frac12
   + \sum_{n=1}^N \frac{z^2}{n^2-z^2}
   + \int_N^{\infty} \frac{z^2}{x^2-z^2} \, dx
   - \frac12 \frac{z^2}{N^2-z^2}
   + O \left( \frac{1}{N^3} \right)
   \tag{3} \\
    \end{align}
$$
as $N \to \infty$, for each $z$.
The right-hand side of (3), without the error term, is what we'll call $f_N(z)$; we can simplify it to
$$
    f_N(z) =
     \sum_{n=1}^N \frac{z^2}{n^2-z^2}
  + z \cdot \frac12 \left(
      \log\left(1+\frac{z}{N}\right)
      - \log\left(1-\frac{z}{N}\right)
  \right)
  - \frac12 \frac{N^2}{N^2-z^2}
  \tag{4}.
$$
I should emphasize that (4) is just a somewhat more elaborate variant of $-\frac12 + \sum_{n=1}^{\infty} \frac{z^2}{n^2-z^2}$, much like how (2) is just a somewhat more elaborate variant of $\sum_{n=1}^{\infty} \frac{1}{n^s}$.
Now when we expand (4) into power series, we recognize the expression from (2) as the coefficients, and we recognize that we can make the sum run from $k=0$ to $\infty$, not just $k=1$ to $\infty$:
$$
    \begin{align}
  f_N(z) &=
   \sum_{n=1}^N \sum_{k=1}^{\infty} \frac{z^{2k}}{n^{2k}}
   + \sum_{k=1}^{\infty} \frac{1}{2k-1} \frac{1}{N^{2k-1}} z^{2k}
   - \frac12 \sum_{k=0}^{\infty} \frac{z^{2k}}{N^{2k}} \\
  &=
      \sum_{k=0}^{\infty} \left(
    \sum_{n=1}^N \frac{1}{n^{2k}}
    + \frac{1}{2k-1} \frac{1}{N^{2k-1}}
    - \frac12 \frac{1}{N^{2k}}
   \right) z^{2k}.
   \tag{5} \\
 \end{align}
$$
This is exactly what we want: take the limit as $N \to \infty$ to conclude that $f(z) = \sum_{k=0}^{\infty} \zeta(2k) z^{2k}$.
(To be rigorous in the final step, we'd need to be more precise about the error term in (1). The actual bound we get from Euler-Maclaurin is
$
    \lvert \mathrm{Error} \rvert
    \le \mathrm{constant} \cdot \int_N^{\infty} \left\lvert \frac{s(s+1)}{x^{s+2}} \right\rvert \, dx 
$
for all $N$ and all $s$ such that $\operatorname{Re}(s) > -1$ and $s \ne 1$.
This lets us control the size of the difference $\sum_{k=0}^{\infty} \zeta(2k) z^{2k} - f_N(z)$.)
This proves that all the even-power coefficients of the power series of $-\frac{\pi z}{2} \cot(\pi z)$ are the corresponding values of $\zeta$, including $\zeta(0)$ as the constant term.
