Advances in Real and Complex Analysis with Applications by Michael Ruzhansky Yeol Je Cho Praveen Agarwal & Iván Area

Author:Michael Ruzhansky, Yeol Je Cho, Praveen Agarwal & Iván Area
Language: eng
Format: epub
Publisher: Springer Singapore, Singapore

Quadratic Reciprocity and Some “Non-differentiable” Functions

Kalyan Chakraborty1 and Azizul Hoque1

(1)Harish-Chandra Research Institute, HBNI, Chhatnag Road, Jhunsi, 211019, Allahabad, India

Kalyan Chakraborty (Corresponding author)

Email: [email protected]

Azizul Hoque

Email: [email protected]

Abstract

Riemann’s non-differentiable function and Gauss’s quadratic reciprocity law have attracted the attention of many researchers. In [28] (Proc Int Conf–Number Theory 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) [12] was the first to give a proof of differentiability/non-differentiability of Riemann’s function. The aim here is to survey some of the work done in these two directions and concentrates more onto a recent work of the first author along with Kanemitsu and Li (Res Number Theory 1, 14, 2015) [5]. In that work (Kanemitsu and Li, Res Number Theory 1, 14, 2015) [5], an integrated form of the theta function was utilised and the advantage of that is that while the theta function 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 behaviour under the increment of the real variable, where the integration is along the horizontal line.

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