Ramanujan tau function on arvau

Wikipedia / Ramanujan tau function

In mathematics, the Ramanujan tau function, studied by Srinivasa Ramanujan, defined by

is the Euler function is the Dedekind eta function is the modular discriminant.

Values

The first few values of the tau function are given in the following table :Calculating this function on an odd square number yields an odd number, whereas for any other number the function yields an even number.

Main properties

, which can be rephrased equivalently as:

is a multiplicative function)
prime.

Together, these two properties are equivalent to the identity

They were proved by Louis Mordell using what is now understood as the theory of Hecke operators., which is called the Ramanujan conjecture. Assuming the first two properties, Ramanujan noted that his conjecture is equivalent to the inequality

is the number-of-divisors function is much easier to show. Ramanujan's conjecture was proved by Pierre Deligne in 1974 as a consequence of his proof of the Weil conjectures.

Ramanujan's L-function

Because the modular discriminant is a cusp form of weight 12, it gives rise to an L-function via the Dirichlet series

above, it has an Euler product

. It satisfies the functional equation

which enables the analytic continuation to all complex numbers, making it an entire function. Ramanujan conjectured that all nontrivial zeros. This is a special case of the Grand Riemann hypothesis. Robert Rankin.

Congruences for the tau function

, the divisor function. Here are some:

, we have

Explicit formulas

In 1972, Ian G. Macdonald proved an explicit formula for the Ramanujan tau functionIn 1975, Douglas Niebur proved the formula

is the sum-of-divisor function.

Conjectures on the tau function

integer newform are integers. Consider the problem:

does not have complex multiplication ?

Indeed, most primes should have this property, and hence they are called ordinary. Despite the big advances by Deligne and Serre on Galois representations are 2, 3, 5, 7, 2411, and .

, an assertion sometimes known as Lehmer's conjecture up to (Apostol 1997, p..