start at the origin, first step number is 1.

- turn $\pi/n$
- move $1/n$ units forward

Angles are cumulative, so this procedure is equivalent (finitely) to $$ u(k):=\sum_{n=1}^{k} \frac{\exp(\pi i H_{n})}{n}$$

- Is the limiting shape formed by a line plot of partial sums a circle or a spiral? Relying on visual intuition with the harmonic numbers seems to be perilous.
- Where's its center? This is equivalent to "If $u(k)$ converges, what does it converge to?".

Naively resumming within exp produces lots of divergent series as a result of the harmonic numbers $H_{n}$. At the cost of being a numerically ill-conditioned sum with lots of cancellation, I have to wonder what this says of arithmetic properties of the harmonic numbers.