Tau-function

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

:$sum\_\{ngeq\; 1\}\; au(n)q^n=qprod\_\{ngeq\; 1\}(1-q^n)^\{24\}.$

The first few values of the tau function are given in the following table OEIS|id=A000594:

If one substitutes $q=exp(2pi\; iz)$ with $zinmathfrak\{h\}=\{z\; in\; mathbb\{C\}\; :\; Im\; z\; >\; 0\}$ then the function $Delta(z):mathfrak\{h\}\; omathbb\{C\}$ defined by

:$Delta(z)=sum\_\{ngeq\; 1\}\; au(n)q^n$

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 $au(n)$:

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

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

Congruences for the tau function

For $kinmathbb\{Z\}$ and $ninmathbb\{Z\}\_\{>0\}$, define $sigma\_k(n)$ 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 $sigma\_k(n)$. Here are some:::::

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For $p\; ot=23$ prime, we have:::