[seqfan] Re: Long email: Complex sum of divisors

[Allan Wechsler]

Since Sven Simon has brought up the subject again: since the spring, I have
used my definition of complex sigma to find about 150 numbers that are
analogous to multiply-perfect numbers using ordinary sigma. Finding
analogues to standard multiply-perfect numbers was, in fact, my main
interest.

I should mention here, though, that it is unclear whether my definition is
really a "sum of divisors" in *any* sense. The trouble is that for powers
of 2 starting at 2^4 = 16, the function counts some divisors twice, due to
the fact that (1+i) and its associates have some real powers. (I think no
other complex Gaussian prime has real powers.)

I kept the definition because it adheres to my desiderata, that the
function applied to real integers is still real, and that the function is
multiplicative. The search for multiply-perfect analogues has been a lot of
fun, so I don't regret it, even though "sum of divisors" may be an
inaccurate description.

On Wed, Sep 21, 2022 at 6:24 AM <sven-h.simon at gmx.de> wrote:

> 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
>
>
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