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# Tau-function

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

$sum_\left\{ngeq 1\right\}tau\left(n\right)q^n=qprod_\left\{ngeq 1\right\}\left(1-q^n\right)^\left\{24\right\}.$

The first few values of the tau function are given in the following table :

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

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

$Delta\left(z\right)=sum_\left\{ngeq 1\right\}tau\left(n\right)q^n$

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 $tau\left(n\right)$:

• $tau\left(mn\right) = tau\left(m\right)tau\left(n\right)$ if $gcd\left(m,n\right) = 1$ (meaning that $tau\left(n\right)$ is a multiplicative function)
• $tau\left(p^\left\{r + 1\right\}\right) = tau\left(p\right)tau\left(p^r\right) - p^\left\{11\right\}tau\left(p^\left\{r - 1\right\}\right)$ for p prime and $rinmathbb\left\{Z\right\}_\left\{>0\right\}$
• $|tau\left(p\right)| leq 2p^\left\{11/2\right\}$ 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 $kinmathbb\left\{Z\right\}$ and $ninmathbb\left\{Z\right\}_\left\{>0\right\}$, define $sigma_k\left(n\right)$ 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 $sigma_k\left(n\right)$. Here are some:

$tau\left(n\right)equivsigma_\left\{11\right\}\left(n\right) bmod 2^\left\{11\right\}mbox\left\{ for \right\}nequiv 1 bmod 8$
$tau\left(n\right)equiv 1217sigma_\left\{11\right\}\left(n\right) bmod 2^\left\{13\right\}mbox\left\{ for \right\} nequiv 3 bmod 8$
$tau\left(n\right)equiv 1537sigma_\left\{11\right\}\left(n\right) bmod 2^\left\{12\right\}mbox\left\{ for \right\}nequiv 5 bmod 8$
$tau\left(n\right)equiv 705sigma_\left\{11\right\}\left(n\right) bmod 2^\left\{14\right\}mbox\left\{ for \right\}nequiv 7 bmod 8$

$tau\left(n\right)equiv n^\left\{-610\right\}sigma_\left\{1231\right\}\left(n\right) bmod 3^\left\{6\right\}mbox\left\{ for \right\}nequiv 1 bmod 3$
$tau\left(n\right)equiv n^\left\{-610\right\}sigma_\left\{1231\right\}\left(n\right) bmod 3^\left\{7\right\}mbox\left\{ for \right\}nequiv 2 bmod 3$

$tau\left(n\right)equiv n^\left\{-30\right\}sigma_\left\{71\right\}\left(n\right) bmod 5^\left\{3\right\}mbox\left\{ for \right\}nnotequiv 0 bmod 5$

$tau\left(n\right)equiv nsigma_\left\{9\right\}\left(n\right) bmod 7mbox\left\{ for \right\}nequiv 0,1,2,4 bmod 7$
$tau\left(n\right)equiv nsigma_\left\{9\right\}\left(n\right) bmod 7^2mbox\left\{ for \right\}nequiv 3,5,6 bmod 7$

$tau\left(n\right)equivsigma_\left\{11\right\}\left(n\right) bmod 691.$

For $pnot=23$ prime, we have

$tau\left(p\right)equiv 0 bmod 23mbox\left\{ if \right\}left\left(frac\left\{p\right\}\left\{23\right\}right\right)=-1$
$tau\left(p\right)equiv sigma_\left\{11\right\}\left(p\right) bmod 23^2mbox\left\{ if \right\} pmbox\left\{ is of the form \right\} a^2+23b^2$
$tau\left(p\right)equiv -1 bmod 23mbox\left\{ otherwise\right\}.$

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