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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>ζ</m:mi> </m:math> ζ function. All the results are based on the complex generalization of the Grünwald-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.