  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/f^{\alpha} spectra of the Fourier transform.
