I wonder whether $\sum\_{k=0}^n \exp(r\_k z)$ has a complex zero for any $n\in \mathbb{Z}\_n^\*,0=r\_0<r\_1<r\_2<\dotsb<r\_n$. I think the answer is affirmative.



Zeros of a complex function
An affirmative answer follows from (9) in [this paper by Ritt](https://www.ams.org/journals/tran/1929-031-04/S0002-9947-1929-1501506-6/S0002-9947-1929-1501506-6.pdf).


