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 $23$,

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