For real $s>1$, the series $\alpha(s):=\underset{n\in\mathbb{N}}{\sum}\frac{1}{n^s}$ converges. The Riemann zeta function $\zeta(s)$ is formed by analytic continuation of $\alpha(s)$ to $\mathbb{C}\smallsetminus\{1\}$. We do this because, in complex analysis, $\alpha(s)$ diverges outside of its defined domain, but I wonder if there is some more arithmetic and less analytical sense by which it extends to (at least some maximal subset of) $\mathbb{C}\smallsetminus\{1\}$.
For example, following Ramanujan's proof, we can compute that $$\alpha(-1)(=1+2+3+\dots)=-\frac{1}{12}$$ using nothing other than a power series and come clever manipulation, even though complex analysis will tell you that series diverges.
I especially wonder about the trivial zeros of $\zeta$. Is there some sense by which it is equally (maybe more) "trivial" that $\alpha(s)=0\ \forall s\in-2\mathbb{N}$. For instance, is there some simple clever proof I've never seen showing that $$1+2+4+9+16+\dots=0?$$
Lastly, I wonder what it is about the harmonic series that makes it so badly behaved. We can find a sense by which (some extension of) these series $\alpha(s)$ converge for every single $s$-value in the complex plane except for $\alpha(1)=1+\frac{1}{2}+\frac{1}{3}+\dots$.