  For the random interval partition of $[0,1]$ generated by the uniform
stick-breaking scheme known as GEM$(1)$, let $u_k$ be the probability that the
first $k$ intervals created by the stick-breaking scheme are also the first $k$
intervals to be discovered in a process of uniform random sampling of points
from $[0,1]$. Then $u_k$ is a renewal sequence. We prove that $u_k$ is a
rational linear combination of the real numbers $1, \zeta(2), \ldots, \zeta(k)$
where $\zeta$ is the Riemann zeta function, and show that $u_k$ has limit $1/3$
as $k \to \infty$. Related results provide probabilistic interpretations of
some multiple zeta values in terms of a Markov chain derived from the interval
partition. This Markov chain has the structure of a weak record chain. Similar
results are given for the GEM$(\theta)$ model, with beta$(1,\theta)$ instead of
uniform stick-breaking factors, and for another more algebraic derivation of
renewal sequences from the Riemann zeta function.
