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Ramanujan sums are exponential sums with exponent defined over theirreducible fractions. Until now, they have been used to provide convergingexpansions to some arithmetical functions appearing in the context of numbertheory. In this paper, we provide an application of Ramanujan sum expansions toperiodic, quasiperiodic and complex time series, as a vital alternative to theFourier transform. The Ramanujan-Fourier spectrum of the Dow Jones index over13 years and of the coronal index of solar activity over 69 years are taken asillustrative examples. Distinct long periods may be discriminated in place ofthe 1/f^{\alpha} spectra of the Fourier transform.

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