A note on the zeroes of the Fredholm series

The issue had been raised whether the Fredholm series $z+z^2+\mathrm{.}\mathrm{.}\mathrm{.}+z^{2^n}+\mathrm{.}\mathrm{.}\mathrm{.}$ has infinitely many zeroes in the unit disk. We provide an affirmative answer, proving that in fact every complex number occurs as a value infinitely many times.