Approximaths

Classic Summation Axioms

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So far, we’ve seen several ways of summing series (the usual method, Euler, Borel, generic, Borel-Écalle and Zeta summation). All of these methods fulfill all three properties — except Zeta summation, which fulfills none of them.

\displaystyle \mathcal{S}(a_0+a_1+a_2+ ...) = \lim_{n \to \infty} \sum_{k =0}^{n}a_k \quad\text{(Regularity)}
\displaystyle \mathcal{S}(\sum(\alpha a_n) + \sum(\beta b_n)) = \alpha \mathcal{S}(\sum a_n) + \beta \mathcal{S}(\sum b_n) \quad\text{(Linearity)}
\displaystyle \mathcal{S}(a_0+a_1+a_2+ ...) = a_0 + \mathcal{S}(a_1+a_2+ ...) \quad\text{(Stability)}

Concerning the Zeta summation we should restrain our enthusiasm by saying that a certain meromorphic complex function called \zeta which have the value \zeta(s)=\sum_{n=1}^{\infty} \frac{1}{n^s} for all Re(s)>1 can be defined on the whole complex plane except at 1, in such a way that, \zeta(0) = -\frac{1}{2}. This ‘summation’ does not rely on any of the properties cited above. So it’s important to be clear about the method we are using and what properties it fulfills.

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