Ramanujan tau function

The Ramanujan tau function is the arithmetic function $\tau :\mathbb{N}\to \mathbb{Z}$ such that, for all $q\in ℂ$ with ,

$$q\prod _{k=1}^{\mathrm{\infty }}{(1-q^k)}^{24}=\sum _{n=1}^{\mathrm{\infty }}\tau (n)q^n.$$

Thus, the Ramanujan tau function is the generating function for the Weierstrass $\mathrm{\Delta }$ function (http://planetmath.org/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

$$q\prod _{k=1}^{\mathrm{\infty }}{(1-q^k)}^{24}.$$

Note that we only need to consider $k=1$ and $k=2$, since higher values of $k$ yield powers (http://planetmath.org/Power) of $q$ that are too large. Thus:

$q{(1-q)}^{24}{(1-q^2)}^{24}$ $=q(1-24q+276q^2-\mathrm{\dots })(1-24q^2+\mathrm{\dots })$

$=q(1-24q+276q^2-\mathrm{\dots }-24q^2+576q^3-\mathrm{\dots })$

$=q(1-24q+252q^2-\mathrm{\dots })$

$=q-24q^2+252q^3-\mathrm{\dots }$

Hence, $\tau (1)=1$, $\tau (2)=-24$, and $\tau (3)=252$.

The sequence $\{\tau (n)\}$ appears in the OEIS as sequence http://www.research.att.com/ njas/sequences/A000594A000594.

Although the values of $|\tau (n)|$ seem to increase rapidly as $n$ increases, the conjecture that $\tau (n)\ne 0$ for all $n\in \mathbb{N}$ has not yet been proven. This conjecture is known as Lehmer’s conjecture.

The Ramanujan tau function has the following properties:

• It is a multiplicative function: For $a,b\in \mathbb{N}$ with $\gcd (a,b)=1$, we have $\tau (ab)=\tau (a)\tau (b)$.

• For any prime $p$ and any $n\in \mathbb{N}$,

  $$\tau (p^{n+1})=\tau (p)\tau (p^n)-p^{11}\tau (p^{n-1}).$$

• For any prime $p$,

  $$|\tau (p)|\le 2p^{\frac{11}{2}}.$$

Ramanujan asserted that $\tau$ several congruences, all of which have been proven. Some simpler examples of such congruences include:

• For any $n\in \mathbb{N}$,

  $$\tau (5n)\equiv 0(mod5).$$

• For any $n\in \mathbb{N}$ and for any nonnegative integer which is a quadratic residue modulo $7$,

  $$\tau (7n-r)\equiv 0(mod7).$$

• For any $n\in \mathbb{N}$ and for any nonnegative integer which is a quadratic residue modulo $23$,

  $$\tau (23n-r)\equiv 0(mod23).$$

Title	Ramanujan tau function
Canonical name	RamanujanTauFunction
Date of creation	2013-03-22 17:51:24
Last modified on	2013-03-22 17:51:24
Owner	Wkbj79 (1863)
Last modified by	Wkbj79 (1863)
Numerical id	12
Author	Wkbj79 (1863)
Entry type	Definition
Classification	msc 11F11
Classification	msc 11A25
Synonym	Ramanujan’s tau function
Related topic	ModularForms
Related topic	ModularDiscriminant
Related topic	Ramanujan
Related topic	ApplicationsOfSecondOrderRecurrenceRelationFormula
Defines	Lehmer’s conjecture