Arithmetical Fourier and limit values of elliptic modular functions

Proceedings of the Indian Academy of Sciences. Mathematical sciences

NIANLIANG WANG^1¹ 

[1] School of Applied Mathematics and Computers, Shangluo University, Shangluo 726000, Shaanxi, People’s Republic of China^1

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

DOI  :  

学科分类：数学（综合）

来源: Indian Academy of Sciences

【 摘 要 】

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.

【 授权许可】

CC BY