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?
Best Answer
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.)