Ramanujan Sums as Derivatives

TL;DR

This paper introduces a Ramanujan operator inspired by Ramanujan sums, demonstrating its derivative-like properties, and presents new methods for computing these sums with applications in signal processing.

Contribution

The paper defines a novel Ramanujan operator with derivative properties and offers a new approach to compute Ramanujan sums using interpolation techniques.

Findings

The Ramanujan operator exhibits first and second derivative properties.

A generalized multiplicative property of Ramanujan sums is established.

A new interpolation-based method for computing Ramanujan sums is developed.

Abstract

In 1918 S. Ramanujan defined a family of trigonometric sum now known as Ramanujan sums. In the last few years, Ramanujan sums have inspired the signal processing community. In this paper, we have defined an operator termed here as Ramanujan operator. In this paper it has been proved that these operator possesses properties of first derivative and second derivative with a particular shift. Generalised multiplicative property and new method of computing Ramanujan sums are also derived in terms of interpolation.