• Arithmetical Fourier and limit values of elliptic modular functions

    • Fulltext

       

        Click here to view fulltext PDF


      Permanent link:
      https://www.ias.ac.in/article/fulltext/pmsc/128/03/0028

    • Keywords

       

      Elliptic modular function; Dedekind eta function; trigonometric series; Dirichlet–Abel theorem; Riemann’s posthumous fragment II

    • Abstract

       

      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.

    • Author Affiliations

       

      NIANLIANG WANG1

      1. School of Applied Mathematics and Computers, Shangluo University, Shangluo 726000, Shaanxi, People’s Republic of China
    • Dates

       
  • Proceedings – Mathematical Sciences | News

    • Editorial Note on Continuous Article Publication

      Posted on July 25, 2019

      Click here for Editorial Note on CAP Mode

© 2021-2022 Indian Academy of Sciences, Bengaluru.