## Resolution of the Quinn-Rand-Strogatz constant of nonlinear physics.(English)Zbl 1200.11098

In an analysis of coupled Winfree oscillators, D. Quinn, R. Rand, and S. Strogatz [Phys. Rev. E 75, 036218-1-10 (2007)] developed a certain $$N$$-oscillator scenario whose bifurcation phase offset $$\phi=\phi_N$$ is implicitly defined, with a conjectured asymptotic behavior $$s_N=\sin\phi_N \sim 1 - c_1/N$$; here $$s=s_N$$ satisfies the algebraic equation $\sum_{n=0}^{N-1}\biggl(2\sqrt{1-s^2(1-2n/(N-1))^2} -\frac1{\sqrt{1-s^2(1-2n/(N-1))^2}}\biggr)=0.$ The article under review is aimed at proving this observation and even the general asymptotic formula $s_N=1-\frac{c_1}N-\frac{c_2}{N^2}-\frac{c_3}{N^3}-\dotsb \qquad\text{as}\quad N\to\infty,$ but also at developing computational and theoretic techniques for estimating the (QRS) constants $$c_1=0.605443657\dots$$ and $$c_2=-0.104685459\dots$$ . For example, $$c_1$$ happens to be a unique real zero of the Hurwitz zeta function $$f(z)=\zeta(1/2,z/2)$$ on the interval $$0<z<2$$. Another curious constant $$C$$ from the Quinn–Rand–Strogatz paper is rigorously identified with $$-f'(c_1)$$.

### MSC:

 11Y60 Evaluation of number-theoretic constants 11M41 Other Dirichlet series and zeta functions 11Z05 Miscellaneous applications of number theory

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