[seqfan] Long email: Complex sum of divisors

[sven-h.simon at gmx.de]

Complex sum of divisors

Some months ago Allan Wechsler wrote an email about a definition of the
complex sum of divisors he used to find multiperfect complex numbers. I did
not want to start a discussion about the topic then, because it is really
complex and I did not want to react to quickly. I do not think that it is
necessary to define a universal complex sigma but we could exchange aspects
of it. Sorry for the lenghty email.

There is a definition by Spira, which is widely used. In his article he
explains in detail, that there are many ways one could define such a
function. Allan Wechsler mentions, that with his definition some divisors
just cancelled out and he got an different result of sigma. (His prime
factorization uses primes with real part bigger than the absolute value of
the imaginary part). In Spira's definition some divisors cancel out too, as
they move around through the quadrants in complex plane. It is just another
mixture of divisors and their associatives. The resulting sigmas are not
connected by just a factor of i, -1 or -i, as the divisors sum up different.

I think Spira lacks a definition of sigma in the different quadrants. Other
than with the positive values of natural numbers one can not avoid divisors
and results of other quadrants
when applying Spiras sigma to a complex number.
My private idea is, that sigma(a+bi) should not be the same value as
sigma(-a-bi). And so my private definition of sigma(a+bi) uses (as Spira)
the prime factorization of a+bi with primes from first quadrant and
(extension to Spira)  the additional factor (1), i, -1, -i that is necessary
in the multiplication of these primes to end up in exactly a+bi. PARI lists
this factor too when factoring a complex number. I apply this factor (1), i,
-1 or -i  on the resulting sigma (Spira) too.

One point came to my attention recently: The result of complex sigma could
be defined to be the same as sigma of natural numbers when applied to primes
4n+3 which remain unchanged as primes in complex plane. This would rule out
a lot of possible definitions (but not the one of Allan Wechsler).

It is not well defined how to handle the function f(n) = sigma(n)-n  - as
used in amicable numbers - in complex plane. When the multiplication of all
prime factors of a+bi is an associative (which was used in the calculation
of sigma) and not a+bi, how calculate sigma(a+bi)-(a+bi) ? Should one use
just a+bi, the associative from first quadrant or the result of the prime
factor multiplication ?

Sven