The Riemann Non-Differentiable Function and Identities for the Gaussian Sums
Abstract
Riemann's example of a continuous, non-differentiable function is given by the summation sin(n2x)/n2. This function is sufficiently irregular and its graph is fractal. Hardy proved that Riemann's non-differentiable function is not differentiable at any irrational point because of a square root singularity at these points. Careful investigation of the differentiability of Riemann's non-differentiable function was carried out by Gerver, who showed that this function has derivative equal to -1/2 at every rational point of a special type (forming the orbit of the point 1 under the theta-modular group). Different proofs of this surprising fact was given by other authors, providing also a close relation between Riemann's non-differentiable function and classical theta-function and Gauss sums. Duistermaat obtained an exact functional equations for this function under transformations of the theta-modular group. In this article we use functional equations for Riemann's non-differentiable function under theta-modular transformations to derive functional equations on Gauss sums generalizing Genocci-Schaar identity.
Document Details
- Document Type
- Technical Report
- Publication Date
- Jan 01, 2000
- Accession Number
- ADP010922
Entities
People
- B. T. Matkarimov