Open Access

Quadratic reciprocity and Riemann’s non-differentiable function

Research in Number Theory20151:14

https://doi.org/10.1007/s40993-015-0015-5

Received: 8 May 2015

Accepted: 7 July 2015

Published: 28 September 2015

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

Keywords

Quadratic reciprocity Theta-transformation Non–differentiable function