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The set of all integer $\mathbb Z$ is closed respect to the operation of addition, this means if we have two integer numbers $n_1$ and $n_2$, the integer $n_3=n_1+n_2$ is still an integer, so even the sum $S=\sum_{k=1}^nk^p$ with $p\in \mathbb N$ must be an integer number. Now we know that using the zeta function regularization we can obtain: $S_0=\sum_{k=1}^\infty{k^0}=-\frac{1}{2}$ My question is this: is $\mathbb Z$ not closed respect to the addition when we add an infinite number of integers or the sum $S_0$ is not well defined in the set $\mathbb Z$? Any suggestion to solve this problem is appreciated.

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    @Gottfried Helms: Right. You have understood my question.2012-03-06

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Addition is a binary operation. It is defined on pairs of integers and, by induction, on finite sets of integers. It is not defined on infinite sets of integers. Now you can define something on infinite sets of integers that is sorta kinda like addition, but it isn't actually addition, and so there's no reason to expect it to do everything addition does. In particular, there's no reason to expect that the result of doing this sorta kinda addition-like thing will result in integers.

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    Actually, this is the same case as putting negative number into gamma function. You can't find out this answer from the original definition because it is not defined. But it is 'created' by analytic continuation.2013-07-17