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::$$au(n)equivsigma_{11}(n) mod 2^{11}mbox{ for }nequiv 1 mod 8:$$au(n)equiv 1217sigma_{11}(n) mod 2^{13}mbox{ for } nequiv 3 mod 8:$$au(n)equiv 1537sigma_{11}(n) mod 2^{12}mbox{ for }nequiv 5 mod 8:$$au(n)equiv 705sigma_{11}(n) mod 2^{14}mbox{ for }nequiv 7 mod 8

:$$au(n)equiv n^{-610}sigma_{1231}(n) mod 3^{6}mbox{ for }nequiv 1 mod 3:$$au(n)equiv n^{-610}sigma_{1231}(n) mod 3^{7}mbox{ for }nequiv 2 mod 3

:$$au(n)equiv n^{-30}sigma_{71}(n) mod 5^{3}mbox{ for }n otequiv 0 mod 5

:$$au(n)equiv nsigma_{9}(n) mod 7mbox{ for }nequiv 0,1,2,4 mod 7:$$au(n)equiv nsigma_{9}(n) mod 7^2mbox{ for }nequiv 3,5,6 mod 7

:$$au(n)equivsigma_{11}(n) mod 691.

For $$p ot=23 prime, we have:$$au(p)equiv 0 mod 23mbox{ if }left(frac{p}{23} ight)=-1:$$au(p)equiv sigma_{11}(p) mod 23^2mbox{ if } pmbox{ is of the form } a^2+23b^2:$$au(p)equiv -1 mod 23mbox{ otherwise}.