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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(p^{r + 1}) = τ(p)τ(p_{r}) − p^{11}τ(p^{r − 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 σ_{k}(n) as the sum of the kth powers of the divisors of n. The tau functions satisfies several congruence relations; many of them can be expressed in terms of σ_{k}(n). Here are some:
For prime, we have Retrieved from "http://en.wikipedia.org/"

