Ramanujan Sums as Derivatives
  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. 
Devendra Kumar Yadav, Gajraj Kuldeep and S. D. Joshi
math.GM