Is it $|y| = \arccos(\frac{e^{-x}}{2})$ the curve described by $h(t) = \sum_{n=1}^\infty \frac{e^{i2\pi\frac{n}{t}}}{n}$?

Question

Let $$ h(t) = \sum_{n=1}^\infty \frac{e^{i2\pi\frac{n}{t}}}{n} $$

A few facts and conjectures that I consider interesting about $h$:

• $h$ gathers well-known results and appears to generalize them:

  • $h(1)$ is the harmonic series. It diverges;

  • $h(2)$ is the alternating harmonic series, with a minus sign. It converges to $-\log{2}$;

  • $h(6)$ appears to converge to $i\frac{\pi}{3}$

• $h$ has functional equations:

  • for any $k \in \mathbb{Z}$, $ h(t) = h(\frac{t}{kt+1}) $;

  • for any $\frac{p}{q} \in \mathbb{Q} - \{ 0 \} $, $ h(\frac{p}{q}) = h(\frac{p}{q \mod p})$;

My question is: is the following conjecture true? "The image of $h$ is the curve with equation $ |y| = \arccos(\frac{e^{-x}}{2}) $", where $x$ is the real part, $y$ the imaginary part.