|
|
|
Viewing Version
6
of
'Ramanujan tau function'
|
[ view 'Ramanujan tau function'
|
back to history
]
| Title of object: |
Ramanujan tau function |
| Canonical Name: |
RamanujanTauFunction |
| Type: |
Definition |
| Created on: |
2008-02-26 01:23:02 |
| Modified on: |
2008-02-26 01:50:10 |
| Classification: |
msc:11F11, msc:11A25 |
| Synonyms: |
Ramanujan tau function=Ramanujan's tau function |
Preamble:
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{pstricks}
\usepackage{psfrag}
\usepackage{graphicx}
\usepackage{amsthm}
\usepackage{xypic}
|
Content:
The \emph{Ramanujan tau function} is the arithmetic function $\tau\colon\mathbb{N}\to\mathbb{Z}$ such that, for all $q\in\mathbb{C}$ with $|q|<1$,
\[
q\prod_{k=1}^{\infty}(1-q^k)^{24}=\sum_{n=1}^{\infty} \tau(n)q^n.
\]
Thus, the Ramanujan tau function is the generating function for the \PMlinkname{Weierstrass $\Delta$ function}{ModularForms}.
Determining values of the Ramanujan tau function directly can be somewhat involved. For example, the values of $\tau(1)$, $\tau(2)$, and $\tau(3)$ will be determined:
To determine $\tau(1)$, $\tau(2)$, and $\tau(3)$, we need to find the coefficient of $q$, $q^2$, and $q^3$, respectively, of the expression
\[
q\prod_{k=1}^{\infty}(1-q^k)^{24}.
\]
Note that we only need to consider $k=1$ and $k=2$, since higher values of $k$ yield \PMlinkname{powers}{Power} of $q$ that are too large. Thus:
\begin{align*}
q(1-q)^{24}(1-q^2)^{24} & =q(1-24q+276q^2-\dots)(1-24q^2+\dots) \\
& =q(1-24q+276q^2-\dots-24q^2+576q^3-\dots) \\
& =q(1-24q+252q^2-\dots) \\
& =q-24q^2+252q^3-\dots
\end{align*}
Hence, $\tau(1)=1$, $\tau(2)=-24$, and $\tau(3)=252$.
The Ramanujan tau function has the following properties:
\begin{itemize}
\item It is a multiplicative function: For $a,b\in\mathbb{N}$ with $\gcd(a,b)=1$, we have $\tau(ab)=\tau(a)\tau(b)$.
\item For any prime $p$ and any $n\in\mathbb{N}$,
\[
\tau(p^{n+1})=\tau(p)\tau(p^n)-p^{11}\tau(p^{n-1}).
\]
\item For any prime $p$,
\[
|\tau(p)|\le 2p^{\frac{11}{2}}.
\]
\end{itemize}
Although the values of $|\tau(n)|$ seem to increase rapidly as $n$ increases, the conjecture that $\tau(n)\neq 0$ for all $n\in\mathbb{N}$ has not yet been proven.
Ramanujan asserted that $\tau$ \PMlinkescapetext{satisfies} several congruences, all of which have been proven. Some simpler examples of such congruences include:
\begin{itemize}
\item For any $n\in\mathbb{N}$,
\[
\tau(5n)\equiv 0\pmod 5.
\]
\item For any $n\in\mathbb{N}$ and for any nonnegative integer $r<7$ which is a quadratic residue modulo $7$,
\[
\tau(7n-r)\equiv 0\pmod 7.
\]
\item For any $n\in\mathbb{N}$ and for any nonnegative integer $r<23$ which is a quadratic residue modulo $23$,
\[
\tau(23n-r)\equiv 0\pmod{23}.
\]
\end{itemize}
\begin{thebibliography}{9}
\bibitem{berndt} Berndt, Bruce C. \emph{Number Theory in the Spirit of Ramanujan}. Providence, RI: American Mathematical Society, 2006.
\end{thebibliography} |
|
|
|
|
|
