Ramanujan tau function

Values

The first few values of the tau function are given in the following table (sequence A000594 in the OEIS):

Ramanujan's conjectures

Ramanujan (1916) observed, but did not prove, the following three properties of τ ( n ) :

The first two properties were proved by Mordell (1917) and the third one, called the Ramanujan conjecture, was proved by Deligne in 1974 as a consequence of his proof of the Weil conjectures (specifically, he deduced it by applying them to a Kuga-Sato variety).

Congruences for the tau function

For k ∈ Z and n ∈ Z_>0, define σ_k(n) as the sum of the k-th powers of the divisors of n. The tau function satisfies several congruence relations; many of them can be expressed in terms of σ_k(n). Here are some:

1. τ ( n ) ≡ σ 11 ( n )   mod   2 11  for  n ≡ 1   mod   8
2. τ ( n ) ≡ 1217 σ 11 ( n )   mod   2 13  for  n ≡ 3   mod   8
3. τ ( n ) ≡ 1537 σ 11 ( n )   mod   2 12  for  n ≡ 5   mod   8
4. τ ( n ) ≡ 705 σ 11 ( n )   mod   2 14  for  n ≡ 7   mod   8
5. τ ( n ) ≡ n − 610 σ 1231 ( n )   mod   3 6  for  n ≡ 1   mod   3
6. τ ( n ) ≡ n − 610 σ 1231 ( n )   mod   3 7  for  n ≡ 2   mod   3
7. τ ( n ) ≡ n − 30 σ 71 ( n )   mod   5 3  for  n ≢ 0   mod   5
8. τ ( n ) ≡ n σ 9 ( n )   mod   7  for  n ≡ 0 , 1 , 2 , 4   mod   7
9. τ ( n ) ≡ n σ 9 ( n )   mod   7 2  for  n ≡ 3 , 5 , 6   mod   7
10. τ ( n ) ≡ σ 11 ( n )   mod   691.

For p ≠ 23 prime, we have

1. τ ( p ) ≡ 0   mod   23  if  ( p 23 ) = − 1
2. τ ( p ) ≡ σ 11 ( p )   mod   23 2  if  p  is of the form  a 2 + 23 b 2
3. τ ( p ) ≡ − 1   mod   23  otherwise .

Conjectures on τ(n)

Suppose that f is a weight k integer newform and the Fourier coefficients a ( n ) are integers. Consider the problem: If f does not have complex multiplication, prove that almost all primes p have the property that a ( p ) ≠ 0 mod p . Indeed, most primes should have this property, and hence they are called ordinary. Despite the big advances by Deligne and Serre on Galois representations, which determine a ( n ) mod p for n coprime to p , we do not have any clue as to how to compute a ( p ) mod p . The only theorem in this regard is Elkies' famous result for modular elliptic curves, which indeed guarantees that there are infinitely many primes p for which a ( p ) = 0 , which in turn is obviously 0 mod p . We do not know any examples of non-CM f with weight > 2 for which a ( p ) ≠ 0 mod p for infinitely many primes p (although it should be true for almost all p ). We also do not know any examples where a ( p ) = 0 mod p for infinitely many p . Some people had begun to doubt whether a ( p ) = 0 mod p indeed for infinitely many p . As evidence, many provided Ramanujan's τ ( p ) (case of weight 12 ). The largest known p for which τ ( p ) = 0 mod p is p = 7758337633 . The only solutions to the equation τ ( p ) ≡ 0 mod p are p = 2 , 3 , 5 , 7 , 2411 , and 7758337633 up to 10 10 .

Lehmer (1947) conjectured that τ ( n ) ≠ 0 for all n , an assertion sometimes known as Lehmer's conjecture. Lehmer verified the conjecture for n < 214928639999 (Apostol 1997, p. 22). The following table summarizes progress on finding successively larger values of N for which this condition holds for all n ≤ N .