Time Measurements, 1/f Noise of the Oscillators and Algebraic Numbers

Abstract

Many complex systems from physics, biology, society… exhibit a 1/f power spectrum in their time variability so that it is tempting to regard 1/f noise as a unifying principle in the study of time. The principle may be useful in reconciling two opposite views of time, the cyclic and the linear one, the philosophic view of eternity as opposed to that of time and death. The temporal experience of such complex systems may only be obtained thanks to clocks which are continuously or occasionally slaved. Here time is discrete with a unit equal to the averaging time of each experience. Its structure is reflected into the measured arithmetical sequence. They are resets in the frequencies and couplings of the clocks, like in any human made calendar. The statistics of the resets shows about constant variability whatever the averaging time: this is characteristic of the flicker (1/f) noise. In a number of electronic experiments we related the variability in the oscillators to number theory, and time to prime numbers. In such a context, time (and 1/f noise) has to do with Riemann hypothesis that all zeros of the Riemann zeta function are located on the critical line, a mathematical conjecture still open after 150 years.