Difference between revisions of "Ramanujan tau"
From specialfunctionswiki
| Line 8: | Line 8: | ||
</gallery> | </gallery> | ||
</div> | </div> | ||
| − | |||
| − | |||
=Properties= | =Properties= | ||
| − | + | [[Ramanujan tau is multiplicative]]<br /> | |
| − | + | [[Ramanujan tau of a power]]<br /> | |
| − | + | [[Ramanujan tau inequality]]<br /> | |
| − | |||
| − | </ | ||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | </ | ||
| − | + | =References= | |
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
[[Category:SpecialFunction]] | [[Category:SpecialFunction]] | ||
Revision as of 00:49, 23 December 2016
The Ramanujan tau function $\tau \colon \mathbb{N} \rightarrow \mathbb{Z}$ is defined by the formulas $$\displaystyle\sum_{n=1}^{\infty} \tau(n)q^n = q \prod_{n=1}^{\infty} (1-q^n)^{24} = \eta(z)^{24}=\Delta(z),$$ where $q=e^{2\pi i z}$ with $\mathrm{Re}(z)>0$, $\eta$ denotes the Dedekind eta function, and $\Delta$ denotes the discriminant modular form.
Plot of $\tau(n)$ for $n=0,1,\ldots,250$.
Properties
Ramanujan tau is multiplicative
Ramanujan tau of a power
Ramanujan tau inequality
