Quadratic reciprocity and Some "non-differentiable" functions

TL;DR

This paper surveys the connections between Riemann's non-differentiable function and Gauss's quadratic reciprocity law, highlighting recent work involving integrated theta functions that extend analysis to the real line.

Contribution

It reviews existing proofs of quadratic reciprocity and differentiability properties of Riemann's function, emphasizing a new approach using integrated theta functions that broadens the analytical framework.

Findings

Integrated theta function $F(z)$ extends analysis to the real line.

Recent work demonstrates new methods for studying non-differentiability.

Survey consolidates classical and recent results on these mathematical topics.

Abstract

Riemann's non-differentiable function and Gauss's quadratic reciprocity law have attracted the attention of many researchers. In \cite{RM} Murty and Pacelli gave an instructive proof of the quadratic reciprocity via the theta-transformation formula and Gerver \cite{G1} was the first to give a proof of differentiability/non-differentiability of Riemnan's function. The aim here is to survey some of the work done in these two questions and concentrates more onto a recent work of the first author along with Kanemitsu and Li \cite{K1}. In that work \cite{K1} an integrated form of the theta function was utilised and the advantage of that is that while the theta-function $\Theta(\tau)$ 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.