• Corpus ID: 123469736

Selfsimilarity of "Riemann's nondifferentiable function"

@inproceedings{Duistermaat1994SelfsimilarityO,
  title={Selfsimilarity of "Riemann's nondifferentiable function"},
  author={Johannes Jisse Duistermaat},
  year={1994}
}
This is an expository article about the series f(x) = 1 X n=1 1 n 2 sin(n 2 x); which according to Weierstrass was presented by Riemann as an example of a continuous function without a derivative. An explanation is given of innitely many selfsimilarities of the graph, from which the known results about the dierentiability properties of f(x) are obtained as a consequence. 
No Paper Link Available
44 Citations
Highly Influential Citations
10
Background Citations
23
Methods Citations
15
Results Citations
3

44 Citations

Some geometric properties of Riemann's non-differentiable function
On the Hausdorff dimension of Riemann’s non-differentiable function
Recent findings show that the classical Riemann's non-differentiable function has a physical and geometric nature as the irregular trajectory of a polygonal vortex filament driven by the binormal…
Simple Proofs of Nowhere-Differentiability for Weierstrass’s Function and Cases of Slow Growth
Using a few basics from integration theory, a short proof of nowhere-differentiability of Weierstrass functions is given. Restated in terms of the Fourier transformation, the method consists in…
The pointwise behavior of Riemann's function
as an example of a function which is continuous but nowhere differentiable. In 1916, Hardy [6] proved, based on earlier work by him and Littlewood [7], that Riemann’s function f is not differentiable…
Analytic and Diophantine properties of certain arithmetic Fourier series
We consider certain Fourier series which arise from modular or automorphicforms. We study their analytic properties: differentiability, modulus of continuity and theH¨older regularity exponent. We…
Asymptotic behaviour and Hausdorff dimension of Riemann's non-differentiable function
Riemann's non-differentiable function, whose analytic regularity has been widely studied, can also be analysed from a geometric perspective. Indeed, it can be generalised to the complex plane to…
Multifractal behavior of polynomial Fourier series
Geometric differentiability of Riemann's non-differentiable function
The Highly Oscillatory Behavior of Automorphic Distributions for SL(2)
Automorphic distributions for SL(2) arise as boundary values of modular forms and, in a more subtle manner, from Maass forms. In the case of modular forms of weight one or of Maass forms, the…
Quadratic reciprocity and Riemann’s non-differentiable function
AbstractRiemann’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.…
...
...

References

SHOWING 1-10 OF 21 REFERENCES
The Differentiability of Riemann's Function
The functiong(x)= :P%-= (sin 7rp2X/7rp2), thought by Riemann to be nowhere differentiable, is shown to be differentiable only at rational points expressible as the ratio of odd integers. The proof…
Modular Functions In Analytic Number Theory
Knopp's engaging book presents an introduction to modular functions in number theory by concentrating on two modular functions, $\eta(\tau)$ and $\vartheta(\tau)$, and their applications to two…
Elementary Number Theory
Designed for a first course in number theory with minimal prerequisites, the book is designed to stimulates curiosity about numbers and their properties. Includes almost a thousand imaginative…
Some problems of diophantine approximation
Riemann’s example of a continuous, ’nondifferentiable’ function
Weierstrass’s non-differentiable function
Riemann’s example of a continuous “nondifferentiable” function continued
Differentiability of Riemann's function
Versuch einer Classification der willkürlichen Functionen reeller Argumente nach ihren Aenderungen in den kleinsten Intervallen.
The Differentiability of the Riemann Function at Certain Rational Multiples of π
...
...