Complex number: equation I would like an hint to solve this equation: $\forall n\geq 1$
$$\sum_{k=0}^{2^n-1}e^{itk}=\prod_{k=1}^{n}\{1+e^{it2^{k-1}}\} \qquad \forall t \in \mathbb{R}.$$
I went for induction but without to much success; I will keep trying, but if you have an hint...
Many thanks.

Let us consider the polynomial $P = \sum_{k=0}^{2^n -1 } X^k $ and $P_2 = \prod_{k=1}^{n} (1+X^{2^{k-1}})$. There exist  a full combinatorial proff of this result. Hint : if you develop the second term and examine all power of $X$ you could find all the binary development of integer between $0$ and $2^{n}-1$. If you want that I complete the proof, ask me. 
