Exercise 5, § 3

Let $(z_n)$ be an infinite sequence of non-zero complex numbers such that $\lim_{n \to \infty} z_n = 1$. Show that, if there exist permutations $\sigma$ of $\mathbf{N}$ such that the infinite product whose general factor is $z_{\sigma(n)}$ is convergent, then the set of products

corresponding to all these permutations can be (i) a single point; or (ii) the whole group $\mathbf{C}^*$; or (iii) an open ray with origin o; or (iv) a circle with centre o; or (v) a "logarithmic spiral", the image of $\mathbf{R}$ under the mapping $t \to a^{t-t_0} (\cos t + i \sin t)$, where $a$ is a real number > 0 and $\neq 1$. (Argue as in Exercise 2 of Chapter VII, § 3, using the fact that the multiplicative group $\mathbf{C}^*$ is isomorphic to the additive group $\mathbf{R} \times \mathbf{T}$.)