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

TL;DR

This paper explores the use of Ramanujan sums as an alternative to Fourier analysis for examining long-period and 1/f noise in time series, demonstrated on financial and solar activity data.

Contribution

It introduces Ramanujan sum expansions for analyzing complex time series, offering a new approach to identify long periods and spectral characteristics.

Findings

Ramanujan sums can distinguish long periods in time series.

The method effectively analyzes 1/f^{} spectra.

Applications to financial and solar data illustrate its utility.

Abstract

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.