Ramanujan tau function

The 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 Weierstrass $\Delta$ 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

 $q\prod_{k=1}^{\infty}(1-q^{k})^{24}.$

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:

 $\displaystyle q(1-q)^{24}(1-q^{2})^{24}$ $\displaystyle=q(1-24q+276q^{2}-\dots)(1-24q^{2}+\dots)$ $\displaystyle=q(1-24q+276q^{2}-\dots-24q^{2}+576q^{3}-\dots)$ $\displaystyle=q(1-24q+252q^{2}-\dots)$ $\displaystyle=q-24q^{2}+252q^{3}-\dots$

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 $\tau(n)\neq 0$ 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 $p$,

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

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

• For any $n\in\mathbb{N}$,

 $\tau(5n)\equiv 0\pmod{5}.$
• For any $n\in\mathbb{N}$ and for any nonnegative integer $r<7$ which is a quadratic residue modulo $7$,

 $\tau(7n-r)\equiv 0\pmod{7}.$
• For any $n\in\mathbb{N}$ and for any nonnegative integer $r<23$ which is a quadratic residue modulo $23$,

 $\tau(23n-r)\equiv 0\pmod{23}.$

References

 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