The Ramanujan tau function, studied by , is the function defined by the following identity: where with , is the Euler function, is the Dedekind eta function, and the function is a holomorphic cusp form of weight 12 and level 1, known as the discriminant modular form (some authors, notably Apostol, write instead of ). It appears in connection to an "error term" involved in counting the number of ways of expressing an integer as a sum of 24 squares. A formula due to Ian G. Macdonald was given in . Values The first few values of the tau function are given in the following table : Ramanujan's conjectures observed, but did not prove, the following three properties of : if (meaning that is a multiplicative function) for prime and . for all primes . 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 (specifically, he deduced it by applying them to a Kuga-Sato variety). Congruences for the tau function For and , define as the sum of the th powers of the divisors of . The tau function satisfies several congruence relations; many of them can be expressed in terms of . Here are some: For prime, we have <li> <li> <li> Explicit formula In 1975 Douglas Niebur proved an explicit formula for the Ramanujan tau function: This also shows that the tau function is always an integer. Conjectures on τ(n) Suppose that is a weight- integer newform and the Fourier coefficients are integers. Consider the problem: Given that does not have complex multiplication, do almost all primes have the property that ? 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 for coprime to , it is unclear how to compute . The only theorem in this regard is Elkies' famous result for modular elliptic curves, which guarantees that there are infinitely many primes such that , which thus are congruent to 0 modulo . There are no known examples of non-CM with weight greater than 2 for which for infinitely many primes (although it should be true for almost all ). There are also no known examples with for infinitely many . Some researchers had begun to doubt