Wikipedia states that Ramanujan sums and the Riemann Zeta function have the same values for even $k$:
$$1 + 2^{2k} + 3^{2k} + \cdots = 0\ (\Re)$$
However, I don't understand how this can be true, because when $k = 0$, because that gives us:
$$1 + 1 + 1 + \cdots = \sum_{n=1}^\infty n^{0} = 0\ (\Re)$$
and yet we have $\zeta(0) = -1/2$, which are clearly unequal.
What am I doing wrong?