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 [14] developed a certain N -oscillator scenario whose bifurcation phase offset φ is implicitly defined, with a conjectured asymptotic behavior: sinφ ∼ 1 − c1/N , with experimental estimate c1 = 0.6054...