This article http://blogs.scientificamerican.com/roots-of-unity/does-123-really-equal-112/ got me thinking about the "identity"
$$1 + 2 + 3 + \cdots = -1/12,$$
and I wanted to convince myself there was nothing particularly unique about this identity or the Riemann zeta construction. More precisely, this identity only really makes sense if you think of an integer $n$ as being the specialization at $z=-1$ of the function $n^{-z}$. So here's a question:
For any complex number $c$, does there exist a domain $\Omega \subset \mathbb{C}$, and analytic functions $F(n, s)_{n\in\mathbb{N}}$ and $f(s)$ on $\Omega$, such that the following hold
i. $F(n,0) = n$
ii. $\sum_{n=1}^\infty F(n,s) = f(s)$ on $\Omega$ in some reasonable sense (maybe converges uniformly on compact subsets of $\Omega$?)
iii. $f$ can be extended holomorphically to some domain containing both $\Omega$ and $0$ such that $f(0) = c$.
So shifting the Euler series and Riemann zeta would be such a construction for $c=-1/12$. As the question stands, I feel that the answer is almost certainly yes, although to be fair the functions $n^{-s}$ have a lot more structure than "holomorphic functions on some domain". So a follow-up question would be: are there "natural" additional constraints for which the answer to this question is No?
I apologize that this is kind of open-ended, but the goal is to convince myself that there is nothing particularly canonical about $-1/12$ (or to hear an explanation of why it is canonical).
Note that $1+2+3+\ldots = (1+1+1+\ldots)+(0+1+2+3+\ldots)$; so if you can regularize $1+1+1+\ldots$ into something nonzero, then you can shift your result away from $-1/12$. Specifically, try $$ 1+2+3+\ldots=\sum_{n=1}^{\infty}\left(n^{z} + (n-1)^{z+1}\right)\big\vert_{z=0}=\zeta(-z)+\zeta(-z-1)\big\vert_{z=0}=\zeta(0)+\zeta(-1)=-\frac{7}{12}, $$ where the sum converges for $\Re (z) < -2$.