The Ramanujan tau function is the function defined by the following identity:
The first few values of the tau function are given in the following table (sequence A000594 in OEIS):
If one substitutes q = exp(2πiz) with then the function defined by
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(mn) = τ(m)τ(n) if gcd(m,n) = 1 (meaning that τ(n) is a multiplicative function)
* τ(pr + 1) = τ(p)τ(pr) − p11τ(pr − 1) for p prime and
* 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 } and , define σk(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 σk(n). Here are some:
For prime, we have