SOME RECENT APPLICATIONS OF THE KERNEL FUNCTION OF GENERALIZED WEYL FRACTIONAL INTEGRALS

Abstract

Quite recently, connections of unusual type have been discussed by the author between the so-called 'fractional calculus' as a new branch of analysis and strong summation processes, furthermore, between fractional integration and certain number theoretic approximation methods. In the following, two different aspects of these inherences are considered: I. a new verification for the powerful method of (D)-summation in case of trigonometric series is given; II. such a generalization of the famous Franel theorem on Riemann's hypo- thesis (1924) is presented which shows the deeper background of the topic in the field of Diophantine approximations.