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This is an extended response to Yiannis Galidakis's question. I have been experimenting with the value $c$ they gave in the previous post. It is in case 3b since $|t| = |W(-\ln c)| = 1$ and there is no $n \in \mathbb{N}$ such that $t^n = 1$. I have evaluated the sequence $a_n$ for $0 \leq n \leq 10^8$. As in Gottfried Helm's analysis, I have found no ...

2

This is not a new answer, but only intended to give some illustration for the cases, that $|b|=1$ and the two subcases, that 1) $b$ is a rational-order complex root of the unit $b = \exp( 2 \pi î /q)$ where $q \in \mathbb Q$ and 2) $b$ is an irrational-order complex root of the unit. Remark: the examples are computed using Pari/GP with ...

5

$\DeclareMathOperator{\Arg}{Arg}$ Let me try to explain what happens with these sequences using a simpler example. Modulo the magnitude of the complex numbers they shuffle, they are almost identical behaviorally to the sequence $a_n=\exp(i n)$, $n\in \mathbb{N}$, for example. Let's see that sequence on the complex plane for say, $N=250$. restart; ...

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