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Zeta function (ζ) is an important function in mathematics and has been discussed by mathematicians in the field of complex analysis, number theory analysis and applied mathematics that are related with the fractional calculus. One’s is an important relation between fractional derivative of the ζ-Riemann and ζ-Hurwitz functions. Fractional derivatives of ζ-Riemann and ζ-Hurwitz functions can be obtained using the Caputo-Ortiguera fractional derivative formula and employe generalizations of the Caputo-Ortiguera fractional derivative from Dirichlet series. An illustration of the Caputo-Ortiguera fractional derivative of ζ-Hurwitz function show on the Gaussian plane to indicate that there are unique characteristics that can give new perspectives on dynamic system problems. One of the illustration discussed is the chaotic of fractional derivatives of Caputo-Ortiguera ζ-Hurwitz function neighborhood of 0.

ζ –Riemann functions, ζ -Hurwitz functions, Caputo-Ortguera Fractional derivative