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?