On Riemann's posthumous fragment II on the limit values of elliptic modular functions

We shall resurrect the instinctive direction of B. Riemann on his posthumous fragment on the limit values of elliptic modular functions A la C.G.J. Jacobi, Fundamenta Nova. In the spirit of Riemann who considered the odd part, we shall realize the situation where there is no singularity occurring in taking the radial limits, thus streamlining and elucidating the recent investigation by Arias de Reyna. By the new Dirichlet-Abel theorem (which should be within reach of Riemann), we may directly sum the series in question, which allows us to condense Arias de Reyna's paper into a few pages.