Quadratic reciprocity and Riemann’s non-differentiable function
© The Author(s) 2015
Received: 8 May 2015
Accepted: 7 July 2015
Published: 28 September 2015
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