Tau-function

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The Ramanujan tau function is the function $\tau:\mathbb{N}\to\mathbb{Z}$ defined by the following identity:

$\sum_{n\geq 1}\tau(n)q^n=q\prod_{n\geq 1}(1-q^n)^{24}.$

The first few values of the tau function are given in the following table (sequence A000594 in OEIS):

$n$	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16
$τ(n)$	1	−24	252	−1472	4830	−6048	−16744	84480	−113643	−115920	534612	−370944	−577738	401856	1217160	987136

If one substitutes $q = exp(2π i z)$ with $z\in\mathfrak{h}=\{z \in \mathbb{C} : \Im z > 0\}$ then the function $\Delta(z):\mathfrak{h}\to\mathbb{C}$ defined by

$\Delta(z)=\sum_{n\geq 1}\tau(n)q^n$

is a holomorphic cusp form of weight 12 and level 1, known as the discriminant modular form.

Ramanujan observed, but could not prove, the following three properties of $τ(n)$ :

$τ(m n) = τ(m)τ(n)$ if $gcd(m, n) = 1$ (meaning that $τ(n)$ is a multiplicative function)
$τ(p r + 1) = τ(p)τ(p r) - p 11 τ(p r - 1)$ for p prime and $r\in\mathbb{Z}_{>0}$
$|\tau(p)| \leq 2p^{11/2}$ for all primes p.

The first two properties were proved by Mordell in 1917 and the third one was proved by Deligne in 1974.

[edit] Congruences for the tau function

For $k\in\mathbb{Z}$ and $n\in\mathbb{Z}_{>0}$ , define $σ k (n)$ as the sum of the $k$ -th powers of the divisors of $n$ . The tau functions satisfies several congruence relations; many of them can be expressed in terms of $σ k (n)$ . Here are some: