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This question was inspired by another recent question (here) .

That older question asked somehow vaguely if the expression $\prod_{k=1}^m \tan(\frac{k\pi}{4m})$ can be simplified (for $m=45$). I propose to formalize this question as follows.

Let $U_n$ denote the set of all complex $n$-th roots of unity, and $U=\bigcup_{n\geq 0}U_n$. Also, put

$$ \begin{array}{c} L_n=\lbrace u_1+u_2+u_3+ \ldots +u_n | u_1,u_2, \ldots ,u_n \in U \rbrace, \\ L=\bigcup_{n\geq 0}L_n \end{array} $$

The additive complexity $c_{add}(l)$ of a number $l \in L$ is the smallest $n$ such that $l\in L_n$. Define

$$ \begin{array}{c} M_n=\lbrace x | x=l_1^{e_1}l_2^{e_2}l_3^{e_3} \ldots l_r^{e_r}, \ \text{with} \ e_i=\pm 1, l_i \in L (1\leq i \leq r), \sum_{k=1}^r c_{add}(l_k)\leq n \rbrace,\\ M=\bigcup_{n\geq 0}M_n \end{array} $$ The additive-multiplicative complexity $c(m)$ of a number $m \in M$ is the smallest $n$ such that $m\in M_n$.

For example, the additive-multiplicative complexity of $t_n=\prod_{k=1}^m \tan(\frac{k\pi}{4m})$ is at most $3+4m$, since

$$ t_n=i^{(-m) \ {\sf mod}\ 4}\prod_{k=1}^{m}\frac{\zeta^k-\zeta^{-k}}{\zeta^k+\zeta^{-k}} \ \ (\text{where} \ \zeta=e^{i\frac{k\pi}{4m}}) $$

Several questions can then be asked :

  • Does the complexity of $t_n$ tend to infinity with $n$ ?

  • What is the complexity of $t_{45}$ ? If this is too hard, can some close lower/upper bounds be found ?

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