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

Abstract

We shall resurrect the instinctive direction of B. Riemann on his posthumous fragment on the limit values of elliptic modular functions à 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.

Limiting Values and Functional and Difference Equations

“…For the latter, as has been noticed [34], there are many instances of radial limits in which the odd part, the first periodic Bernoulli polynomial, of the polylogarithm function of order 1 appears as a result of eliminating the real part, the log sin function. For examples, see [38,39]. The polylogarithm function of order 1 is indeed the monologarithm function, the ordinary logarithm function extended to the circle of convergence (|z| = 1, z = 1)…”

“…We elucidate the main ingredients from the first author's paper [38] and its sequel [39]. Reference [38] condenses the 67 pages long paper [40] into 17 pages.…”

“…Theorem 1 ( [38], Theorem 3). Let ξ = M Q be a rational number with M even and Q > 1 and let z = ye πiξ , y ∈ [0, 1).…”

“…To state the Dirichlet-Abel theorem ( [38], Theorem 1) we need rudiments of the Discrete Fourier Transform (DFT). Its theory is stated in many literature, [35,36], ( [41], pp.…”

Limiting Values and Functional and Difference Equations

“…We elucidate the main ingredients from the first author's paper [86] and its sequel [87]. [86] condenses the 67 pages long paper [21] into 17 pages.…”