Ramanujan tau function

[Image:Absolute Tau function for x up to 16,000 with logarithmic scale.JPG|thumbnail|upright=1.64|Values of
for with a logarithmic scale. The blue line picks only the values of that are multiples of 121.]
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. 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 :

	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

Calculating this function on an odd square number yields an odd number, whereas for any other number the function yields an even number.

Ramanujan's conjectures

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

if
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.

Congruences for the tau function

For and, the Divisor function is 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

Explicit formula

In 1975 Douglas Niebur proved an explicit formula for the Ramanujan tau function:
where is the sum of the positive divisors of.

Conjectures on the tau function

Suppose that is a weight- integer newform and the Fourier coefficients are integers. Consider the problem:
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 . The only solutions up to to the equation are 2, 3, 5, 7, 2411, and .
conjectured that for all, an assertion sometimes known as Lehmer's conjecture. Lehmer verified the conjecture for up to . The following table summarizes progress on finding successively larger values of for which this condition holds for all.

	reference
	Lehmer
	Lehmer
	Serre, Serre
	Jennings
	Jordan and Kelly
	Bosman
	Zeng and Yin
	Derickx, van Hoeij, and Zeng

Ramanujan's L-function

Ramanujan's -function is defined by
if and by analytic continuation otherwise. It satisfies the functional equation
and has the Euler product
Ramanujan conjectured that all nontrivial zeros of have real part equal to.