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Spinning_Sandwich comments on Rationality Quotes September 2012 - Less Wrong
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That's like saying the standard choice of branch cut for the complex logarithm is arbitrary.
And?
When you complexify, things get messier. My point is that making a generalization is possible (though it's probably best to sum over integers with 0 \leq arg(z) < \pi, as you pointed out), which is the only claim I'm interested in disputing. Whether it's nice to look at is irrelevant to whether it's functional enough to be punnable.
You're right -- the generalization works.
Mainly what I don't like about it is that \sigma(z) no longer has the nice properties it had over the integers: for example, it's no longer multiplicative. This doesn't stop Gaussian integers from being friendly, though, and the rest is a matter of aesthetics.