# Natural extension of the divisor function $\sigma$ to the complex integers $\mathbb{Z}[i]$

The divisor function $\sigma$ of a positive integer $n$ is defined as the sum of the positive divisors of $n$. It turns out that this definition is equivalent to

$$\sigma(n)=\prod_{\substack{p^\alpha||n\\p\in\mathbb{P}}}\frac{p^{\alpha+1}-1}{p-1}\tag{1}$$

We can extend this function to the non-zero integers by simply neglecting the negative divisors. What we actually do, mathematically, is making the factorization unique - this is possible only if we have one unit. As before, we could actually use the expression $(1)$ to define $\sigma$.

Now, I wonder how the divisor function could be naturally extended to the non-zero complex integers $\mathbb{Z}[i]^*$. There are several obvious candidates, but all of them fail in one way or another. I will describe the perhaps most natural.

Were we to sum over all divisors of a complex integer $z$, it is obvious that the divisors would cancel and the sum would be zero; this was also the case for the negative integers. We avoided this by finding a unique factorization of every integer.

This time, the same idea cannot be applied directly, but what we can do is write any non-zero complex integer as the product of complex primes in the first quadrant of the complex plane and a unit. We then shave off the unit, and use the expression $(1)$ for the complex prime powers; that is,

$$\sigma(z)=\prod_{p^\alpha||z}\frac{p^{\alpha+1}-1}{p-1}\tag{2},$$

where each $p$ is a complex prime lying in the first quadrant.

Unfortunately, this function will fail to correspond to the original for real integers; what is more, it is even complex-valued there. I find this drawback important, as it means that this extended function cannot directly be used to find new properties of the classical one (which is often the case for complex extension).

One might wonder if there is any use of a invented function like this. However, there is also a good part; the function is still multiplicative and it satisfies the inequality $|\sigma(z)|\ge z$. Also, Robert Spira already studied this function in 1961.

My question is, what is the most meaningful extension that still corresponds to the original divisor function for the real integers?

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Don't sum divisors, but norms of divisors, with divisors considered up to unit multiple. Remember that all ideals in ${\mathbf Z}[i]$ are principal. For nonzero ideals $(\alpha)$, define $\sigma((\alpha))$ to be the sum of ${\rm N}(\delta)$, where $(\delta)$ runs over the principal ideals generated by factors of $(\alpha)$. For example, if $(\alpha)$ is a prime ideal then $\sigma((\alpha)) = 1 + {\rm N}(\alpha)$. This summatory function is multiplicative and has the same type of product formula as in ${\mathbf Z}$, but in place of Gaussian integer primes in the formula you have their norms. – KCd May 12 '13 at 1:35
As a reason this definition is nice, classically we have $\sum_{n \geq 1} \sigma(n)/n^s = \zeta(s)\zeta(s-1)$, where $\zeta(s)$ is the Riemann zeta-function, and $\sum_{(\alpha)} \sigma((\alpha))/{\rm N}(\alpha)^s = \zeta_{{\mathbf Q}(i)}(s)\zeta_{{\mathbf Q}(i)}(s-1)$, where $\zeta_{{\mathbf Q}(i)}(s)$ is the zeta-function of ${\mathbf Q}(i)$. – KCd May 12 '13 at 1:38
@Tharsis, the following 2007 REU paper from Auburn University might interest you. – Kashitokiku Teshikiari Oct 14 '13 at 6:44
This paper by McDaniel might also be relevant for your purposes @Tharsis. – Kashitokiku Teshikiari Oct 14 '13 at 6:52
@Librecoin, you might also want to peruse the papers of Colin Defant. – Kashitokiku Teshikiari Jan 18 '15 at 22:23