When you're dealing with arithmetic functions, you might have come across the classical Möbius' function
$$ \mu(n)=\begin{cases} (-1)^{\omega(n)}=(-1)^{\Omega(n)} &\mbox{if }\; \omega(n) = \Omega(n)\\ 0&\mbox{if }\;\omega(n) < \Omega(n).\end{cases}, $$ where $ω(n)$ is the number of distinct primes dividing the number $n$ and $Ω(n)$ is the number of prime factors of $n$, counted with multiplicities.
Is there a complex analogon $\mu^{\Bbb C}(z)$ that additionally takes into account that primes of the form $z=4n+1$ might be factored as well, e.g. $5=(1+2i)(1-2i)$?
So a number $z_n$ that contains a prime of that form would give
- $\mu^{\Bbb C}(z_n)=0$, when think of e.g. $5=(1+2i)(1-2i)$ as a square
- or $\mu^{\Bbb C}(z_n)=1$ when think of it as product of two Gaussian primes.
A more general question, but I'm not sure how this is related, is: How does the concept of factoring natural numbers carry over to complex natural numbers?
Anything to read on that topic would be nice...
There is slight trouble with choice of unit, using ideals instead of Gaussian primes as factors resolves this.
As always the Möbius function encodes the inclusion-exclusion principle.
To invert $$F(\mathfrak n) = (f*1)(\mathfrak n) = \sum_{\mathfrak d|\mathfrak n} f(\mathfrak d)$$ we will use $$\mu(\mathfrak p_1^{r_1} \mathfrak p_2^{r_2} \cdots \mathfrak p_k^{r_k}) = \begin{cases} (-1)^{k} \; \text{ when each $r_i=1$} \\ 0 \; \text{ otherwise}\end{cases}.$$
Then simplifying a bit $$(F*\mu)(\mathfrak p_1^{r_1} \mathfrak p_2^{r_2} \cdots \mathfrak p_k^{r_k}) = \sum_{\epsilon = \{0,1\}^k} (-1)^{\epsilon_1 + \epsilon_2 + \cdots + \epsilon_r} F(\mathfrak p_1^{r_1-\epsilon_1} \mathfrak p_2^{r_2-\epsilon_2} \cdots \mathfrak p_k^{r_k-\epsilon_k}) = f(\mathfrak p_1^{r_1} \mathfrak p_2^{r_2} \cdots \mathfrak p_k^{r_k}).$$
It may be worth checking that we get the same correspondence with Dirichlet series/zeta functions as in the natural numbers case.