Herein we develop connections between zeta functions and some
recent ``mysterious'' constants of nonlinear physics.
In an important analysis of coupled
Winfree oscillators, Quinn, Rand, and Strogatz developed a certain
$N$-oscillator scenario whose bifurcation phase offset $\phi$ is
implicitly defined, with a conjectured asymptotic behavior $\sin
\phi \sim 1 - c_1/N$, with experimental estimate $c_1 =
0.605443657\dotsc$. We are able to derive the exact theoretical value of
this ``QRS constant'' $c_1$ as a real zero of a particular
Hurwitz zeta function. This discovery enables, for example,
the rapid resolution of $c_1$ to extreme precision. Results and
conjectures are provided in regard to higher-order terms of the
$\sin \phi$ asymptotic, and to yet more physics constants emerging from the
original QRS work.