Nulls are located at all primes on the real axis?

Question

This page claims "a complex function, whose nulls are located at all primes on the real axis"

$$f(z) = \left| 2 - \sum\limits^\infty_{k=1} \frac{1}{k} \frac{e^{2 \pi i z}-1}{ e^{2 \pi i z / k}-1} \right|$$

Does this formula have a name? Is the claim true?

Answer 1

It's a trick!

Hint: Which values of $z$ make

$$ e^{2\pi i z} - 1 = 0 $$

and

$$ e^{2\pi i z/k} - 1 \neq 0 $$

for all integers $k \geq 2$, $k \neq z$?