Abstract
Riemann’s non-differentiable function and Gauss’s quadratic reciprocity law have attracted the attention of many researchers. Here we provide a combined proof of both the facts. In (Proc. Int. Conf.–NT 1;107–116, 2004) Murty and Pacelli gave an instructive proof of the quadratic reciprocity via the theta-transformation formula and Gerver (Amer. J. Math. 92;33–55, 1970) was the first to give a proof of differentiability/non-differentiabilty of Riemnan’s function. We use an integrated form of the theta function and the advantage of that is that while the theta-function Θ(τ) is a dweller in the upper-half plane, its integrated form F(z) is a dweller in the extended upper half-plane including the real line, thus making it possible to consider the behavior under the increment of the real variable, where the integration is along the horizontal line.
2010 Mathematics Subject Classification: Primary: 11A15, Secondary: 11F27
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Butzer, PL, Stark, EL: “Riemann’s example” of a continuous nondifferentiable function in the light of two letters (1865) of Christoffel to Prym. Bull. Soc. Math. Belg. Sèr. A. 38, 45–73 (1986).
Cauchy, AL: Methode simple et nouvelle pour la determination complete des sommes alternee, formees avec les racines primitives des equattions binomes. J. Math. Pure Appl. (Liouvilee). 5, 154–183 (1840).
Duistermaat, JJ: Self-similarity of “Riemann’s nondifferentiable function. Nieuw Arch. Wisk. 9, 303–337 (1991).
Gerver, J: The differentiability of the Riemann function at certain rational multiples of π. Amer. J. Math. 92, 33–55 (1970).
Gerver, J: More on the differentiability of the Riemann function. Amer. J. Math. 93, 33–41 (1970).
Gerver, J: On cubic lacunary Fourier series. Trans. Amer. Math. Soc. 355, 4297–4347 (2003).
Hardy, GH: Weierstrass non-differentiable functions. Trans. Amer. Math. Soc. 17, 301–325 (1916).
Holschneider, M, Tchamichian, P: Pointwise analysis of Riemann’s nondifferentiable function. Invent. Math. 105, 157–275 (1991).
Itatsu, S: Differentiability of Riemann’s function. Proc. Japan Acad. Ser. A Math. Sci. 57, 492–495 (1981).
Jaffard, S: The spectrum of sigularities of Riemann’s function. Rev. Mat. Iberoamericana. 12, 441–460 (1996).
Kahane, JP: Lacunary Taylor and Fourier series. Trans. Amer. Math. Soc. 79, 199–213 (1964).
Kanemitsu, S, Tsukada, H: Contributions to the theory of zeta-functions—modular relation supremacy. World Scientific, London (2014).
Meyer, Y: Le traitment du signal et l’analyze mathématique. Ann. Inst. Fourier Grenoble. 50, 593–632 (1994).
Murty, R, Pacelli, A: Quadratic reciprocity via theta functions. Proc. Int. Conf.–Number Theory No. 1, 107–116 (2004).
Neuenschwander, E: Riemann’s example of a continuous “non-differentiable” function. The Mathematical Intelligencer. 1(1), 40–44 (1978).
Segal, SL: Riemann’s example of a continuous “non-differentiable” function continued. Math. Intell. 1(2), 81–82 (1978).
Smith, A: The differentiability of Riemann’s function. Proc. Amer. Math. Soc. 34, 463–468 (1983).
Ulrich, P: Anmerkungen zum “Riemannschen Beispiel” einer stetigen, nicht differenzierbaren Funktion. Res. Math. 31, 245–265 (1997).
Acknowledgement
The authors are indebted to the anonymous referee for his(her) enlightening comments which enabled us to make the paper more orderly (specially the ‘Introduction’).
The second author is supported in part by JSPS Grant in-aid for Scientific Research No. 17540050
To Professor Kálmán Györy on his seventy-fifth birthday with great respect and friendship.
Author information
Authors and Affiliations
Corresponding author
Additional information
Competing interests
The authors declare that they have no competing interests.
Rights and permissions
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.
About this article
Cite this article
Chakraborty, K., Kanemitsu, S. & Long, L.H. Quadratic reciprocity and Riemann’s non-differentiable function. Res. number theory 1, 14 (2015). https://doi.org/10.1007/s40993-015-0015-5
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s40993-015-0015-5