Fractional derivative of the Hurwitz ζ-function and chaotic decay to zero

In this paper the fractional order derivative of a Dirichlet series, Hurwitz zeta function and Riemann zeta function is explicitly computed using the Caputo fractional derivative in the Ortigueira sense. It is observed that the obtained results are a natural generalization of the integer order derivative. Some interesting properties of the fractional derivative of the Riemann zeta function are also investigated to show that there is a chaotic decay to zero (in the Gaussian plane) and a promising expression as a complex power series.