'Syncing' Up with the Quinn-Rand-Strogatz Constant: Hurwitz-Zeta Functions in Non-Linear
This work extends the analytical and computational investigation of the Quinn-Rand-Strogatz (QRS) constants from non-linear physics. The QRS constants (c1, c2, ..., cN) are found in a Winfree oscillator mean-field system used to examine the transition of coupled oscillators as they lose synchronization. The constants are part of an asymptotic expansion of a function related to the oscillator synchronization. Previous work used high-precision software packages to evaluate c1 to 42 decimal-digits, which made it possible to recognize and prove that c1 was the root of a certain Hurwitz-zeta function. This allowed a value of c2 to be conjectured in terms of c1. Therefore there is interest in determining the exact values of these constants to high precision in the hope that general relationships can be established between the constants and the zeta functions. Here, we compute the values of the higher order constants (c3, c4) to more than 42-digit precision by extending an algorithm developed by D.H. Bailey, J.M. Borwein and R.E. Crandall. Several methods for speeding up the computation are explored and an alternate proof that c1 is the root of a Hurwitz-zeta function is attempted.