Summing a divergent series and a constant combined At least according to the answer to  this question, $\zeta(1) = \gamma $ (once reqularized, of course).
Let me rephrase that by stating that:
$$ \sigma(\zeta(1)) = \gamma $$
Here, $\sigma(x)$ is the 'summation-function'. It's a function that assigns a value to any $x$, using Borel, Abel, Ramanujan, Euler, Cesaro or any other summation method that works  (e.g. It makes a divergent series summable). The $\sigma$-function 'chooses' a summation method that suits $x$ best (to assign a (finite) constant to it). We assume that the different summation methods dont have different 'working' values for the same $x$ (I now call upon  this question).
Furthermore, we denote $C$ as a converging series and $D$ as a diverging one.
What would $\sigma(C + D) $ be? Is it $\sigma(C) + \sigma(D)$ ? Or what would, for example,
$\sigma(\zeta(1)^3 + \zeta(2))$ be?  
So, to summarize my question: Could you please explain the properties of the $\sigma$-function to me, with relation to $C$ and $D$ ?
Thanks a lot in advance.
P.S. A bonus question: What do you think of the 'summation-function'? is it useful or just mathematical bogus? Or has it been defined (even more) properly already?
 A: Making sense of "picks a summation method that works" is very difficult, because for many series there are different reasonable choices.  A standard method of summing bad series is "zeta-function regularization" --- for example, the method is popular in physics, because S. Hawking uses it to compute QFT on curved backgrounds.  In its easiest form, let $\sum a_n$ be the series you want to sum: then you can consider the function $\zeta_a(s) = \sum a_n^{-s}$.  When the sequence $a_n$ is positive and grows at least as $n^\epsilon$ for some $\epsilon>0$, then $\zeta_a$ will converge in the far-right part of the complex plane.  Now you can hope that it has a singly-valued analytic continuation to $s = -1$.
However, this summation method will not satisfy the linearity that you want.  One example: you can look up values for zeta functions of the form $\sum (an+b)^{-s}$ and see directly the failure of additivity.

More generally, you should look at Hardy, Divergent Series.  Among other statements in there are some no-go theorems, of the form: there is no function $\{\text{series}\} \to \{\text{numbers}\}$ that agrees with the Cauchy convergence on convergent series and that satisfies some natural requirements.  (Unfortunately, I don't have the book with me, and I don't remember any exact versions of such a theorem.)
A: In fact you could have asked for more. 
Let AC the set of absolutely convergent series, and $S:AC\rightarrow \mathbb{C}$ the $\mathbb{C}$-algebra homomorphism that associates to a convergent series its sum.
Then we may ask for an extension $\sigma$ of $S$, defined on some subalgebra $D_1$ of the set D of all series, that satisfies the following rules.


*

*regularity: if $s\in D_1$ is converging, then $\sigma(s)=S(s)$,

*invariance by translation: $\sigma(\sum_0^\infty 
a_n)=a_0+\sigma(\sum_1^\infty a_n)$,

*linearity: $\sigma$ is $\mathbb{C}$-linear,

*product: $\sigma$ is an homomorphism for multiplication.
Abelian summations methods satisfy these four rules. These methods
associate to a divergent sum $\sum a_n$ a function, say $\sum a_n x^n$, and try to take its value at $x=1$ by some process related to analytic continuation. If you can read French, the first chapter of the book "Series divergentes et theories asymptotiques", by J.P. Ramis, is a nice introduction to these questions. The author surveys resummation methods for divergent series, from Leibniz to Ecalle.
A: Define $\tau(C+D):=\sigma(C)+\sigma(D)$. Then $\tau$ is a summation method for $C+D$, and by your assumption of uniqueness it follows that $\sigma(C+D)=\tau(C+D)=\sigma(C)+\sigma(D)$.
A: I am currently working on a theory that assigns to divergent sums the values from a set of "extended" numbers. Each extended number has a transfinite and standard(or regular) part. The standard part corresponds to the regularized value of the series or integral.
In this theory the regular part function is linear: for extended numbers $w$, $w_1$ and $w_2$ and a regular number $a$ the following holds:
$$\operatorname{reg} (w_1+w_2)=\operatorname{reg}w_1+\operatorname{reg}w_2$$
and
$$\operatorname{reg} (a w)=a\operatorname{reg}w$$
On the other hand, the regular part of product of two transfinite numbers  is not usually a product of their regular parts.
For instance, 
$$\operatorname{reg} \int_0^\infty 1\, dx=0$$
but
$$\operatorname{reg} \left(\int_0^\infty 1\, dx\right)^2=-\frac1{12}$$
$$\operatorname{reg} \sum_{k=1}^\infty 1=-\frac12$$
but 
$$\operatorname{reg} \left(\sum_{k=1}^\infty 1\right)^2=\frac16$$
