The Lévy-Steinitz Theorem asserts that for each sequence $(v_i)_{i\geq 1}$ of vectors in $\mathbb{R}^n$, the set
$$S:=S((v_i)_{i\geq 1}):=\left\{\sum_{i=1}^{\infty} v_{\pi (i)}\in \mathbb{R}^n\:|\: \pi :\mathbb{N}\to \mathbb{N} \text{ bijective} \right\}$$
of convergent rearrangements of the corresponding series is either empty or a translate of some subspace of $\mathbb{R}^n $.
In particular, for a sequence $(z_i)_{i\geq 1}$ in $\mathbb{C}$, we have that $S\left((c_i)_{i\geq 1}\right)$ is either empty, a singleton, all of $\mathbb{C}$ or a line in the complex plane.
Now I was wondering if this can be used to make some assertion about the set convergent rearrangements of a product of complex numbers, i.e. about the set $$ P:=P((z_i)_{i\geq 1}):=\left\{\prod_{i=1}^{\infty} v_{\pi (i)}\in \mathbb{C}\:|\: \pi:\mathbb{N}\to \mathbb{N} \text{ bijective} \right\}\text{.}$$
My thoughts:
In $\mathbb{R}$, given a sequence $(c_i)_{i\geq 1}$, we have (if everything is defined) that $$\prod_{i=1}^{\infty}c_{\pi(i)} = \exp\left(\log\left(\prod_{i=1}^{\infty}c_{\pi(i)}\right)\right)=\exp\left(\sum_{i=1}^{\infty}\log(c_{\pi(i)})\right)\text{.}$$
So (by Riemann's Rearrangement Theorem) one obtains $$P\left((c_i)_{i\geq 1}\right)= \begin{cases} \exp(\sum_{i=1}^{\infty}\log(c_i)) &\text{if } \sum_{i=1}^{\infty}\log(c_i) \text{ converges absolutely} \\ \mathbb{R}^+_0 \text{ or } \mathbb{R}^-_0 &\text{if }\sum_{i=1}^{\infty}\log(c_i) \text{ is conditionally convergent}\\ \emptyset &\text{else}\\ \end{cases}$$
Logarithmising in $\mathbb{C}$ and considering the case where $S(\log(z_i)_{i\geq 1}))$ is a line, say $S=\{a+tb:t\in \mathbb{R}\}$ for some $a,b\in \mathbb{C}$, we would get that $P((z_i)_{i\geq 1}))=\{\exp(a+tb):t\in \mathbb{R}\}$, which describes a spiral in $\mathbb{C}$.
However in $\mathbb{C}$, the logarithm is only unique up to integer multiples of $2\pi i$, so this approach seems rather useless to me.
Any ideas?
The logarithm is fixed one for all. Also $c_j \to 1$ (otherwise the product never converges whatever is the rearrangement) and you'll choose any value of $\log c_j$ such that $\log c_j\to 0$.
Thus the ambiguity is a $2i\pi k_j$ term for finitely many $\log c_j$ which won't change the image by $\exp$ of the obtained subset of $\Bbb{C}\cup-\infty$.
The set you'll obtain will be : either a point, a spiral $e^{a+bt},t\in \Bbb{R}\cup \infty$ for some $ \Re(b)< 0$, a circle $e^{a+it}$, empty, or the whole of $\Bbb{C}$.