Ramanujan sums analysis of long-period sequences and 1/f noise

Michel Planat (1) , Milan Minarovjech (2) , Metod Saniga (2)

1 FEMTO-ST - Franche-Comté Électronique Mécanique, Thermique et Optique - Sciences et Technologies (UMR 6174)
2 Astronomical Institute of the Slovak Academy of Sciences

Ramanujan sums are exponential sums with exponent defined over the irreducible fractions. Until now, they have been used to provide converging expansions to some arithmetical functions appearing in the context of number theory. In this paper, we provide an application of Ramanujan sum expansions to periodic, quasiperiodic and complex time series, as a vital alternative to the Fourier transform. The Ramanujan-Fourier spectrum of the Dow Jones index over 13 years and of the coronal index of solar activity over 69 years are taken as illustrative examples. Distinct long periods may be discriminated in place of the 1/fα spectra of the Fourier transform.

Mots clés: sommes de Ramanujan; traitement du signal

Domaines: Physique mathématique [math-ph]; Physique mathématique [math-ph]; Théorie des nombres [math.NT]

Michel Planat, Milan Minarovjech, Metod Saniga. Ramanujan sums analysis of long-period sequences and 1/f noise. EPL - Europhysics Letters, 2009, 85, pp.40005. ⟨hal-00346442⟩