Fractional Calculus, Zeta Functions And Shannon Entropy

This paper deals with the fractional calculus of zeta functions. In particular, the study is focused on the Hurwitz zeta function. All the results are based on the complex generalization of the Grunwald-Letnikov fractional derivative. We state and prove the functional equation together with an integral representation by Bernoulli numbers. Moreover, we treat an application in terms of Shannon entropy.