This paper deals with the fractional calculus of zeta functions. In particular, the study is focused on the Hurwitz ζ \zeta 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.

中文翻译：

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分数演算，ζ函数和香农熵

本文讨论了zeta函数的分数演算。特别地，研究集中在Hurwitzζ\ zeta函数上。所有结果均基于Grünwald-Letnikov分数导数的复杂概括。我们陈述并证明函数方程以及通过伯努利数的积分表示。此外，我们根据香农熵来处理应用程序。