Gaussian Numbers

[Eugene McDonnell]

On Jan 16, 2005, at 1:29 AM, sven-h.simon wrote:

> I would appreciate clear definitions for functions like Sigma and so 
> on in the
> complex plane too.
> In the discussions, an article of Spira was mentionend, which I 
> ordered at once.
>
> I think complex numbers are an important mathematical component 
> (Fourier
> Transforms, Riemann ... ) and I am little bit surprised about the lack 
> of
> definitions, knowledge and research here. On the other hand the 
> currently
> discussed themes are mainly math fun.
>
> Complex numbers are a complex theme, so I do not think, I should do 
> research
> work in the fundamental concepts here.
>
> One question I have in this context: I contributed several sequence 
> about counts
> of complex primes. These primes were ordered by their norm (was 
> sometimes called
> abs here, I think). I counted 1+i as the first complex prime, and 
> continued with
> primes corresponding to the primes 1 mod 4 .
> The next one is 1 + 2i (with 5 as corresponding prime). Is it common 
> sense to
> count 1+i as the first complex prime ?
>
> Complex primes for me are primes in the complex plane, that have both 
> real and
> imaginary parts as natural integers greater than 0. The other primes 3 
> mod 4
> appear only on the axis of the complex plane, having real or imaginary 
> part 0.
> The imaginary part is just the prime itself multiplicated with the 
> unit i.
>
> Of course it would not bother me, if my definition is not common sense 
> ..., I
> just would like to know. For me starting with 1+i in the sequence of 
> 'complex
> primes' as defined above is not completly wrong.
>
> Sven Simon
> Germany
>

Please keep in mind that I am not a mathematician, so bear with me in 
what follows. I sympathize with Simon, in his surprise about the lack 
of definitions in some key number-theoretical parts of the complex 
numbers. I, too, experienced this more than thirty years ago when I was 
asked to provide a compatible definition of residue for complex 
numbers. I quickly found that residue and floor are closely 
intertwined; that you can't define one without the other. Compatability 
meant that floor and residue would behave for complex numbers just the 
way they did for the reals. However, there was not a compatible 
definition of complex floor. Hurwitz many years before had defined 
complex floor to be: the floor w of complex number z was the complex 
integer w at the center of the unit square centered about w containing 
z -- with some additional remarks about which vertices and sides of 
that square belonged to w. I couldn't use this because it used Gauss's 
definition of least residues, not the least nonnegative residues that 
applied to the reals, and that used the real floor function, which made 
possible the proof of the fundamental theorem of arithmetic.

I came away with compatible definitions of floor and residue which were 
superior to Hurwitz's in that the fundamental theorem applied unaltered 
to complex as well as to real numbers -- because it had the consequence 
that z - floor z was strictly less than 1.

Since compatability with the reals was my goal, I changed my view of 
prime decomposition by restating it to say that a real integer has a 
unique decomposition into real positive primes, thus leaving the way 
open for a comparable definition of the primes for complex integers. 
"Real positive" for the complex numbers means the first quadrant, 
including the positive axis -- but using the first quadrant primes and 
not those real primes with remainder 1 mod 4.

I'd welcome hearing additional views from you -- damnations not 
excluded.

Eugene McDonnell