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Fractional calculus, complex functions and function spaces

Last modified: 2023-05-14

#### Abstract

This paper deals with the fractional calculus of complex functions. More precisely, we study the case of zeta functions. In fact, this family of functions seems to be of independent interest with respect to fractional calculus. For instance, the fractional derivative of zeta functions is given by its integer derivative by simply interchanging integer and fractional derivative orders. The same holds for many other properties of this function family (e.g., functional equation, link with the distribution of prime numbers). Furthermore, the fractional calculus of zeta functions can be extended to Banach algebras by showing their central role in modern fractional calculus.