Ramanujan Sums as Derivatives and Applications

Abstract

In 1918, S. Ramanujan defined a family of trigonometric sums now known as Ramanujan sums. In this letter, we define a class of operators based on the Ramanujan sums termed here as Ramanujan class of operators. We then prove that these operators possess properties of first derivative and with a particular shift, of second derivative also. Applications of Ramanujan class of operators for edge detection and noise level estimation are also demonstrated.