Arithmetical Fourier and limit values of elliptic modular functions

Here, we shall use the first periodic Bernoulli polynomial B¯1(x)=x-[x]-12\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\bar{B}_1(x)=x-[x]-\frac{1}{2}$$\end{document} to resurrect the instinctive direction of B Riemann in his posthumous fragment II on the limit values of elliptic modular functions à la C G J Jacobi, Fundamenta Nova §40 (1829). In the spirit of Riemann who considered the odd part, we use a general Dirichlet–Abel theorem to condense Arias–de-Reyna’s theorems 8–15 into ‘a bigger theorem’ in Sect. 2 by choosing a suitable R-function in taking the radial limits. We supplement Wang (Ramanujan J. 24 (2011) 129–145). Furthermore, the same method is applied to obtain in Sect. 3 a correct representation for the ‘trigonometric series’, i.e., we prove that for every rational number x the trigonometric series (3.5) is represented by ∑n=1∞(-1)nB¯1(nx)n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sum _{n=1}^{\infty }(-1)^n\frac{{\bar{B}}_1(nx)}{n}$$\end{document} as Dedekind suggested but not by ∑n=1∞B¯1(nx)n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sum _{n=1}^{\infty }\frac{{\bar{B}}_1(nx)}{n}$$\end{document} as Riemann stated.