A physical basis to Riemann hypothesis 1/f frequency noise

Planat, Michel
May 2000
AIP Conference Proceedings;2000, Vol. 517 Issue 1, p538
Academic Journal
The paper reports on a possible link between a physical problem as old as electronics: 1/f electrical noise (Schottky, 1926) and a mathematical problem going back to the foundation of analytical number theory: the properties of Riemann zeta function and Riemann hypothesis (1859). The low frequency noise of hyperbolic power spectral density is common to many fields of physics. We first remind basic results concerning electrical noise and how there were related to progress in physics. We then introduce non linearity as a possible source of 1/f noise of oscillators: carrier to envelope resonance in a field effect transistor is reported. The main part of the paper is devoted to diophantine aspects in 1/f frequency noise. Using a set mixer+filter it is shown that the mixing of input oscillators leads to a beat signal of which the frequency is controlled from continued fraction expansions of the frequency ratio of signals at the input. The amplitude of beat signals is defined from the position of fractions with respect to the equally spaced graduation. 1/f frequency noise is observed at the resonance and this is related to finite continued fractions and sums over waves with fractional phases. Finally we remind some properties of primes and of the zeta function in connexion with the 1/f noise of oscillators. © 2000 American Institute of Physics.


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