This paper: http://eprints.maths.ox.ac.uk/741/1/smallbone2.pdf has a lot of interesting things to say about the problem of extending the sum of divisors function to number fields, including a lot of references. Victor On Wed, Jun 8, 2011 at 4:10 PM, Henry Baker <hbaker1@pipeline.com> wrote:
Isn't this problem somewhat akin to defining mod(p,q), p,q complex numbers? The problem is that almost every definition is arbitrary: besides |mod(p,q)|<|q|, and the fact that mod(p+n*q,q)=mod(p,q) (n an integer), and perhaps mod(p+i*n*q,q)=mod(p,q), what other criteria should we use to pin down this function?
We could require the image of mod(p,q) to be a parallelogram in (1,i) space, but that might be far too confining: we might want to use one of Gosper's dragons as the basic "rep-tile".
BTW, I assume that there is some sort of connection between mod(p,q) and elliptic functions? I've never seen such a connection written down, but the basic 2-D periodicity would seem to scream it out.
At 02:04 PM 6/8/2011, Dan Asimov wrote:
I was going to ask the same question.
It seems the canonicalest choice would be elements z of Z[i] with Re(z) > 0 and Im(z) >= 0. But there's something arbitrary about this.
But even more natural might be to define the sum of aliquot parts in something like Z[i] / z ~ iz. I would hope this object would retain some kind of useful algebraic structure, that would be related to the ring structure of Z[i] as Z+ is related to the group structure of Z.
--Dan
Gene wrote:
<< In the ring Z[i], the units (divisors of 1) are +1, -1, +i, -i. Thus if d is a divisor of 3+i, so are -d, id, -id. In taking the sum of divisors, how do you make a canonical choice among the associates of each divisor? If you take them all, the sum is zero. In the rational integers, one chooses the positive divisor.
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