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 §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 \(\sum _{n=1}^{\infty }(-1)^n\frac{{\bar{B}}_1(nx)}{n}\) as Dedekind suggested but not by \(\sum _{n=1}^{\infty }\frac{{\bar{B}}_1(nx)}{n}\) as Riemann stated.
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References
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Acknowledgements
The author would like to express his hearty thanks to Prof. Shigeru Kanemitsu for providing him with the direction of this research and for enlightening discussions. The author would also like to thank Prof. Kalyan Chakraborty for enlightening discussion. This work was supported by Natural Science Basic Research Project of Shaanxi Province of China (Program No. 2016JM1034) and by Shangluo Science Research Plan (Program No. SK2014-01-08) and by Science Research Project of Shaanxi Provincial Department of Education (Program Nos 16JM1265, 16JK1238).
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Wang, N. Arithmetical Fourier and limit values of elliptic modular functions. Proc Math Sci 128, 28 (2018). https://doi.org/10.1007/s12044-018-0408-1
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DOI: https://doi.org/10.1007/s12044-018-0408-1
Keywords
- Elliptic modular function
- Dedekind eta function
- trigonometric series
- Dirichlet–Abel theorem
- Riemann’s posthumous fragment II