Let $P_s(n) = \sum_{j=1}^{n} j^{-s}$. For fixed $s$
near $s=\frac12$, we divide the unit interval into bins and count
how many of the partial sums $P_s(1)$, $P_s(2)$, \dots, $P_s(N)$ lie
in each bin $\mone$.
The properties of the histogram are predicted by a random model
unless $s=\frac12$.
When $s=\frac12$ the histogram
is surprisingly flat, but has a few strong spikes.
To explain the surprises at $s=\frac12$, we use classical
results about Diophantine approxmation, lattice points, and
uniform distribution of sequences.