Articles written in Proceedings – Mathematical Sciences

    • Arithmetical Fourier and limit values of elliptic modular functions


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      Here, we shall use the first periodic Bernoulli polynomial $\bar{B}_{1}(x) = x-[x]-\frac{1}{2}$ 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 $\S$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. Wesupplement 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 $\sum^{\infty}_{n=1}(-1)^{n}\frac{\bar{B}_{1}(nx)}{n}$ as Dedekind suggested but not by $\sum^{\infty}_{n=1}\frac{\bar{B}_{1}(nx)}{n}$ as Riemann stated.

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