"The first quadrant" is well-defined, except possibly concerning what part of the boundary is included. It is what we learned in analytic geometry. I'm not aware that there is a canonical choice among the four associates of a Gaussian integer. The choice suggested here does have one remaining ambiguity: 1+i or 1-i ? Perhaps a better definition for a sum of divisors function would be to use the norm of the divisor rather than the divisor itself. (The norm of an algebraic number is the product of all its conjugates. For imaginary quadratic fields, it's the absolute value squared.) Now it's independent of the choice of associate. This easily generalizes to the sum of the n-th powers of divisors. This also extends to other rings of algebraic integers. When the ring does not possess unique factorization, there seems to be a choice: use only integer divisors, or allow ideals as divisors. -- Gene
________________________________ From: Marc LeBrun <mlb@well.com> To: math-fun <math-fun@mailman.xmission.com> Sent: Thursday, June 9, 2011 1:06 PM Subject: Re: [math-fun] Some Sum
This is interesting, but I'm still having trouble with "THE first quadrant".
Isn't the usual thing to take as the canonical quadrant for Gaussian factors the quarter plane to the right of the origin bounded by x=y and x=-y, rather than the all-non-negative upper-right quadrant bounded by the axes?
The choice impacts what the "sum of divisors" is, right?