Ramanujan tau function



The Ramanujan tau function, studied by, is the function $$\tau : \mathbb{N} \rarr\mathbb{Z}$$ defined by the following identity:
 * $$\sum_{n\geq 1}\tau(n)q^n=q\prod_{n\geq 1}\left(1-q^n\right)^{24} = q\phi(q)^{24} = \eta(z)^{24}=\Delta(z),$$

where $|τ(n)|$ with $n < 16,000$, $$\phi$$ is the Euler function, $n$ is the Dedekind eta function, and the function $q = exp(2πiz)$ is a holomorphic cusp form of weight 12 and level 1, known as the discriminant modular form (some authors, notably Apostol, write $$\Delta/(2\pi)^{12}$$ instead of $$\Delta$$). 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 :

Calculating this function on an odd square number (i.e. a centered octagonal 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 $Im z > 0$:


 * $Δ(z)$ if $τ(n)$ (meaning that $τ(n)$ is a multiplicative function)
 * $τ(mn) = τ(m)τ(n)$ for $η$ prime and $gcd(m,n) = 1$.
 * $τ(n)$ for all primes $n$.

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 $τ(p^{r + 1}) = τ(p)τ(p^{r}) − p^{11} τ(p^{r − 1})$ and $r > 0$, the Divisor function $|τ(p)| ≤ 2p^{11/2}$ is the sum of the $p$th powers of the divisors of $p$. The tau function satisfies several congruence relations; many of them can be expressed in terms of $k ∈ $\mathbb{Z}$$. Here are some:
 * 1) $$\tau(n)\equiv\sigma_{11}(n)\ \bmod\ 2^{11}\text{ for }n\equiv 1\ \bmod\ 8$$
 * 2) $$\tau(n)\equiv 1217 \sigma_{11}(n)\ \bmod\ 2^{13}\text{ for } n\equiv 3\ \bmod\ 8$$
 * 3) $$\tau(n)\equiv 1537 \sigma_{11}(n)\ \bmod\ 2^{12}\text{ for }n\equiv 5\ \bmod\ 8$$
 * 4) $$\tau(n)\equiv 705 \sigma_{11}(n)\ \bmod\ 2^{14}\text{ for }n\equiv 7\ \bmod\ 8$$
 * 5) $$\tau(n)\equiv n^{-610}\sigma_{1231}(n)\ \bmod\ 3^{6}\text{ for }n\equiv 1\ \bmod\ 3$$
 * 6) $$\tau(n)\equiv n^{-610}\sigma_{1231}(n)\ \bmod\ 3^{7}\text{ for }n\equiv 2\ \bmod\ 3$$
 * 7) $$\tau(n)\equiv n^{-30}\sigma_{71}(n)\ \bmod\ 5^{3}\text{ for }n\not\equiv 0\ \bmod\ 5$$
 * 8) $$\tau(n)\equiv n\sigma_{9}(n)\ \bmod\ 7\text{ for }n\equiv 0,1,2,4\ \bmod\ 7$$
 * 9) $$\tau(n)\equiv n\sigma_{9}(n)\ \bmod\ 7^2\text{ for }n\equiv 3,5,6\ \bmod\ 7$$
 * 10) $$\tau(n)\equiv\sigma_{11}(n)\ \bmod\ 691.$$

For $n ∈ $\mathbb{Z}$_{&gt;0}$ prime, we have $$\tau(p)\equiv 0\ \bmod\ 23\text{ if }\left(\frac{p}{23}\right)=-1$$ $$\tau(p)\equiv \sigma_{11}(p)\ \bmod\ 23^2\text{ if } p\text{ is of the form } a^2+23b^2$$ $$\tau(p)\equiv -1\ \bmod\ 23\text{ otherwise}.$$ 

Explicit formula
In 1975 Douglas Niebur proved an explicit formula for the Ramanujan tau function:


 * $$\tau(n)=n^4\sigma(n)-24\sum_{i=1}^{n-1}i^2(35i^2-52in+18n^2)\sigma(i)\sigma(n-i).$$

where $σ_{k}(n)$ is the sum of the positive divisors of $k$.

Conjectures on τ(n)
Suppose that $n$ is a weight-$n$ integer newform and the Fourier coefficients $σ_{k}(n)$ are integers. Consider the problem:
 * Given that $f$ does not have complex multiplication, do almost all primes $k$ have the property that $p ≠ 23$?

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 $&sigma;(n)$ for $f$ coprime to $p$, it is unclear how to compute $a(n)$. The only theorem in this regard is Elkies' famous result for modular elliptic curves, which guarantees that there are infinitely many primes $n$ such that $a(p) ≢ 0 (mod p)$, which thus are congruent to 0 modulo $a(n) (mod p)$. There are no known examples of non-CM $p$ with weight greater than 2 for which $a(p) (mod p)$ for infinitely many primes $p$ (although it should be true for almost all $f$). There are also no known examples with $a(p) = 0$ for infinitely many $p$. Some researchers had begun to doubt whether $p$ for infinitely many $p$. As evidence, many provided Ramanujan's $a(p) ≢ 0 (mod p)$ (case of weight 12). The only solutions up to 1010 to the equation $a(p) ≡ 0 (mod p)$ are 2, 3, 5, 7, 2411, and $p$.

conjectured that $a(p) ≡ 0 (mod p)$ for all $p$, an assertion sometimes known as Lehmer's conjecture. Lehmer verified the conjecture for $7,758,337,633$ up to $n$ (Apostol 1997, p. 22). The following table summarizes progress on finding successively larger values of $n$ for which this condition holds for all $τ(p)$.

Ramanujan's L-function
Ramanujan's L-function is defined by
 * $$L(s)=\sum_{n\ge 1}\frac{\tau (n)}{n^s}$$

if $$\Re s>6$$ and by analytic continuation otherwise. It satisfies the functional equation
 * $$\frac{L(s)\Gamma (s)}{(2\pi)^s}=\frac{L(12-s)\Gamma(12-s)}{(2\pi)^{12-s}},\quad s\notin\mathbb{Z}_0^-, \,12-s\notin\mathbb{Z}_0^{-}$$

and has the Euler product
 * $$L(s)=\prod_{p\,\text{prime}}\frac{1}{1-\tau (p)p^{-s}+p^{11-2s}},\quad \Re s>7.$$

Ramanujan conjectured that all nontrivial zeros of $$L$$ have real part equal to $$6$$.