Ramanujan tau function


The Ramanujan tau function, studied by, is the function defined by the following identity:
where with and 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. 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 :
12345678910111213141516
1−24252−14724830−6048−1674484480−113643−115920534612−370944−5777384018561217160987136

Ramanujan's conjectures

observed, but did not prove, the following three properties of :
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.

Congruences for the tau function

For k ∈ Z and n ∈ Z>0, define σk 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 σk.
Here are some:
For p ≠ 23 prime, we have




Conjectures on ''τ''(''n'')

Suppose that is a weight integer newform and the Fourier coefficients are integers. Consider the problem: If does not have complex multiplication, prove that 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, we do not have any clue as to how to compute. The only theorem in this regard is Elkies' famous result for modular elliptic curves, which indeed guarantees that there are infinitely many primes for which, which in turn is obviously. We do not know any examples of non-CM with weight for which mod for infinitely many primes . We also do not know any examples where mod for infinitely many. Some people had begun to doubt whether indeed for infinitely many. As evidence, many provided Ramanujan's . The largest known for which is. The only solutions to the equation are and up to.
conjectured that for all, an assertion sometimes known as Lehmer's conjecture. Lehmer verified the conjecture for . The following table summarizes progress on finding successively larger values of for which this condition holds for all.
Nreference
3316799Lehmer
214928639999Lehmer
Serre, Serre
1213229187071998Jennings
22689242781695999Jordan and Kelly
22798241520242687999Bosman
982149821766199295999Zeng and Yin
816212624008487344127999Derickx, van Hoeij, and Zeng