Background:
One way to "prove" that $1+2+3+\dots=-\frac{1}{12}$ is to define the function $f(z)=\sum_{n=0}^\infty \frac{1}{n^z}$ for $z\in\mathbb C, \operatorname{Re}(z)>1,$ then define $\zeta(z)$ as the analytic continuation of $f$ to all of $\mathbb C$, and then show that $\zeta(-1)=-\frac{1}{12}$. Naively replacing the terms in the sum defining $f$ with $z=-1$ suggests that $1+2+3+\dots=-\frac{1}{12}$.
I am curious whether this general idea could be used to rigorously define sums of divergent series. Given a formal series such as $1+2+3+\dots$, could one define the sum to be the analytic continuation of any function definable somewhere in $\mathbb C$ using a summation formula that reproduces this sum somewhere else? Would such a definition lead to a unique answer?
Question:
Let $\{a_n\}_{n\in\mathbb N_0}$ be a family of functions $a_n:\mathbb C\to\mathbb C,z\mapsto a_n(z)$ such that:
- $a_n(z_0)=n$ for all $n\in\mathbb N_0$ and for some $z_0\in\mathbb C$,
- $f(z):=\sum_{n=0}^\infty a_n(z)$ is well defined on some open subset $A\subset\mathbb C$.
Let $g$ be the analytic continuation of $f$ to the rest of $\mathbb C$. Must $g(z_0)=-\frac{1}{12}$?
