The Ramanujan tau function, studied by , is the function defined by the following identity:
,
where with and is the Dedekind eta function

Dedekind eta function

The Dedekind eta function, named after Richard Dedekind, is a function defined on the upper half-plane of complex numbers, where the imaginary part is positive...

 and the function is a holomorphic

Holomorphic function

In mathematics, holomorphic functions are the central objects of study in complex analysis. A holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighborhood of every point in its domain...

 cusp form

Cusp form

In number theory, a branch of mathematics, a cusp form is a particular kind of modular form, distinguished in the case of modular forms for the modular group by the vanishing in the Fourier series expansion \Sigma a_n q^n...

 of weight 12 and level 1, known as the discriminant modular form.

Values

The first few values of the tau function are given in the following table :

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

Ramanujan's conjectures

observed, but could not prove, the following three properties of :

• if (meaning that is a multiplicative function
  Multiplicative function
  In number theory, a multiplicative function is an arithmetic function f of the positive integer n with the property that f = 1 and whenevera and b are coprime, then...
  
  )
• for p prime and r>0.
• for all primes
  Prime number
  A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2...
  
   p.

The first two properties were proved by and the third one, called the Ramanujan conjecture, was proved by Deligne in 1974 as a consequence of his proof of the Weil conjectures

Weil conjectures

In mathematics, the Weil conjectures were some highly-influential proposals by on the generating functions derived from counting the number of points on algebraic varieties over finite fields....

.

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 functions satisfies several congruence relations; many of them can be expressed in terms of σ_k(n).
Here are some:

1. 
2. 
3. 
4. 
5. 
6. 
7. 
8. 
9. 
10. 

For p ≠ 23 prime, we have

11. 
    
12. 
    
13. 
    

Conjectures on tau(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 multplication, prove that almost allprimes 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 tau(p) (case of weight 12). The largest known p for which tau(p)=0 mod p is p=7758337633. The only solutions to the equation are p = 2, 3, 5, 7, 2411, and 7758337633 up to

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.

n	reference
3316799	Lehmer (1947)
214928639999	Lehmer (1949)
	Serre (1973, p. 98), Serre (1985)
1213229187071998	Jennings (1993)
22689242781695999	Jordan and Kelly (1999)
22798241520242687999	Bosman (2007)

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