TITLE: Is $1+2+3+4+\cdots=-\frac{1}{12}$ the unique ''value'' of this series?
QUESTION [18 upvotes]: I'm reading about zeta-function regularization in physics and I have some mathematical doubt.  I understand  that, since a sum of infinite terms is not well defined in a field, a series that is considered divergent in the  usual meaning  can have a ''value'', defined in some less conventional way. 
The zeta-function regularization is one of such ways that, as an example, assign the value $-\frac{1}{12}$ to the infinite series $S=1+2+3+4+...$ using the fact that the zeta-function $\zeta(s)$ is the analytic continuation of the series $\sum_{n=1}^\infty n^{-s}$, and $\zeta(-1)=-\frac{1}{12}$.
But how we can be sure that there does not exists other possible ''regularizations'' that gives different values to the same series? 
And, if more than one value does exist, there is some  criterion to select between them?  Or the zeta-function regularization is preferred only for physical motivations (because the experiments confirm its values)?
On the web I've found a lot of posts less or more reliable about this topic, someone knows a reference to a well defined axiomatic approach to the problem of the value of divergent series?

REPLY [5 votes]: I'll go a bit further than Hagen Von Eitzen's (very good) answer. 
I had considered the general case as he does, to see whether $1+2+...= -1/12$ had any "deep" content.
So I'll give here the results I found (which do not, if I remember correctly, require the axiom of choice), and if you are interested I can provide proofs. 
Call admissible space a sub-vector space of $K^{\Bbb{N}}$ for $K=\Bbb{R}, \Bbb{C}$ that contains all summable sequences and is closed under $(a_n)_n \to (0,a_0,a_1,....)$.
If $H$ is an admissible space, and if $T$ is a linear form on $H$, say that $T$ is a supersummation if and only if : 

it extends the sum for summable sequences, and
$T((0,a_0,a_1,\ldots)) = T((a_n)_n)$, ie if appending $0$ to the front of a sequence does not change the ‘sum’.

Say that a supersummation $T$ on the admissible space $H$ is proper if and only if $T$ is the unique supersummation on $H$.
Now one has the following results :

Assume $H$ is an admissible space, with a proper supersummation $T$ and $u$ is a given fixed sequence. Then $H$ is contained in an admissible space containing $u$ with a proper supersummation on it, if and only if $u$ satisfies a linear induction relation, up to an $H$-term. The second condition reads, more precisely: there exist $k\in\Bbb{N}$, $a_0,\ldots,a_{k-1}$ and $h\in H$ such that $u_{n+k} = \displaystyle\sum_{i=0}^{k-1} a_i u_{n+i} + h_n$ for all $n\in\Bbb{N}$.
There exists a (necessarily unique) admissible space $H$ satisfying the following properties: 1. $H$ has a proper supersummation; 2. no admissible space properly containing $H$ has a proper supersummation; moreover 3. every admissible space with a proper supersummation, $(H’,T’)$, is contained in $(H,T)$,
ie, $H’\subseteq H$ und $T’=T\mid H'$.

This second result is very interesting as it can be read ‘If you want to define a generalization of infinite sums, there is only one way to do it right, and the way of doing it is, in a sense, absolute’. And, as Hagen Von Eitzen shows, $H$ cannot contain $(n)_n$ but it does contain $(2^n)_n$, and the value of the supersummation of $(2^n)$ is $-1$.
EDIT: I'm adding to this answer a short "paper" that I had written about this topic. A few remarks : 1. This is in french (because I'm french), so until I trnalsate it (if I translate it), only those who read french can understand it. 2. I wrote it at the beginning of my second year post-highschool, so there are some things that I could have done differently and really more easily -don't be surprised if there are calculations that could be avoided, or trivial arguments that are developed etc. 3. At some point I use Zorn's lemma, but as those who read it can see, it's not necessary. I simply use it to get the existence of a maximal element, but I prove its existence in another way later: I'm only using it to show what the maximal element might look like, so that I can better pursue the search for said element. With this in mind, here's the "paper"