Difference between revisions of "Riemann function"

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$$R(x)=\displaystyle\sum_{k=1}^{\infty} \dfrac{\sin(k^2 x)}{k^2}.$$
 
$$R(x)=\displaystyle\sum_{k=1}^{\infty} \dfrac{\sin(k^2 x)}{k^2}.$$
  
=Properties=
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<strong>Theorem:</strong> The Riemann function is is [[continuous]].
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File:Riemannplot.png|Plot of $R(x)$ on $[0,1]$.
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File:Riemannfunction.gif|The partial sum $R(x,N)=\displaystyle\sum_{k=1}^N \dfrac{\sin(k^2 x)}{k^2}$ for various values of $N$.
<strong>Proof:</strong> █
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=Properties=
<strong>Theorem:</strong> The Riemann function is [[nowhere differentiable]] except at points of the form $\pi \dfrac{2p+1}{2q+1}$ with $p,q \in \mathbb{Z}$.
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[[Riemann function is continuous]]<br />
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[[Riemann function is almost nowhere differentiable]]<br />
<strong>Proof:</strong> █
 
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=References=
 
=References=
 
[https://pure.ltu.se/ws/files/30923977/LTU-EX-03320-SE.pdf]<br />
 
[https://pure.ltu.se/ws/files/30923977/LTU-EX-03320-SE.pdf]<br />
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[[Category:SpecialFunction]]

Latest revision as of 03:26, 6 July 2016

The Riemann function is the function $R \colon \mathbb{R} \rightarrow \mathbb{R}$ defined by $$R(x)=\displaystyle\sum_{k=1}^{\infty} \dfrac{\sin(k^2 x)}{k^2}.$$

Properties

Riemann function is continuous
Riemann function is almost nowhere differentiable

References

[1]