About the limiting cases of the Riemann function

Abstract

Let us investigate the limiting cases of the Riemann function. Let us take the differential equation P of the function and consider the functions (see article). Following Riemann's example, we imagine the path of integration in the form of a flexible, stretchable and easily movable thread. As it moves, the special point deforms this path, pushing it in front of itself and never crossing it. With this representation of the integration path from closed curves corresponding to integrals (11), (13), (15), for integrals (16), (18), (20) we obtain open paths in certain directions extending to infinity.