Ramanujan's tau function

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):

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

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

Number Theory

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