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 OEIS|id=A000594:
If one substitutes 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 :
* if (meaning that is a multiplicative function)
* 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 as the sum of the -th powers of the divisors of .The tau functions satisfies several congruence relations; many of them can be expressed in terms of . Here are some:::::
::
:
::
:
For prime, we have:::