Riemann function

From specialfunctionswiki
Revision as of 22:52, 31 December 2015 by Tom (talk | contribs) (Created page with "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=...")
(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)
Jump to: navigation, search

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

Theorem: The Riemann function is is continuous.

Proof:

Theorem: The Riemann function is nowhere differentiable except at points of the form $\pi \dfrac{2p+1}{2q+1}$ with $p,q \in \mathbb{Z}$.

Proof: