Gaussian Numbers

[Franklin T. Adams-Watters]

The definition I learned way back when is that sigma for Gaussian integers is the sum of the norms of the divisors (where equivalent divisors are only counted once).  Of course, with this definition, perfect Gaussian integer is not a very interesting concept.

While this definition is questionable for sigma, the equivalent definition for phi - that is, the sum_d|n phi(d)=norm(n) - seems to be pretty clearly right, since it gives the number of residues modulo n that are relatively prime to n.

Really, any definition that involves picking one of the equivalent values for each divisor (such as the first quadrant rule) is pretty arbitrary - unless somebody can come up with a rule for choosing an equivalent form U(n) for each Gaussian integer n with the properties that:
 (1) for any n,m, U(n*m) = U(n) * U(m)
 (2) U(n) can be calculated significantly more easily than factoring n.
or at least:
 (2') U(n) is "natural" in some way that does not depend on factoring n.

Granted, 2 is a bit vague, and 2' is very vague; but I know of know candidates for U(n) that don't depend very explicitly on factoring n.

sven-h.simon at t-online.de (sven-h.simon) wrote:
>I would appreciate clear definitions for functions like Sigma and so on in the 
>complex plane too.

-- 
Franklin T. Adams-Watters
16 W. Michigan Ave.
Palatine, IL 60067
847-776-7645


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