The Riemann hypothesis is considered and validated in this paper. For that purpose a multipoint Taylor series is used to obtain a series representation around the four inter-related zeroes. This series allows getting a new equation whose solutions are the searched zeroes. It is shown that this equation only has solutions on the critical line.

The Ambartsumian equation is studied and solved. A simple solution expressed in terms of a linear combination of Mittag-Leffler functions is deduced and implemented. A fast algorithm for its numerical realization is described and examplified.

The substantial, tempered, and shifted fractional derivatives are introduced in a unified framework. Their properties are studied and, in the light of the strict sense criterion for derivative definitions, they are characterized and assessed. In the scope of the framework, new tempered linear sytems and transfer functions are introduced.