Ramanujan tau function

File:Absolute Tau function for x up to 16,000 with logarithmic scale.JPG

Values of $|\tau (n)|$ for $n<16,000$ with a logarithmic scale. The blue line picks only the values of $n$ that are multiples of 121.

The Ramanujan tau function, studied by Ramanujan (1916), is the function $\tau :\mathbb{\mathbb{N}}\to \mathbb{\mathbb{Z}}$ defined by the following identity:

$\sum _{n\ge 1}\tau (n)q^n=q\prod _{n\ge 1}{(1-q^n)}^{24}=q\varphi (q{)}^{24}=\eta (z{)}^{24}=\mathrm{\Delta }(z),$

where $q=\exp (2\pi iz)$ with $\mathrm{I}\mathrm{m}(z)>0$, $\varphi$ is the Euler function, $\eta$ is the Dedekind eta function, and the function $\mathrm{\Delta }(z)$ is a holomorphic cusp form of weight 12 and level 1, known as the discriminant modular form (some authors, notably Apostol, write $\mathrm{\Delta }/(2\pi {)}^{12}$ instead of $\mathrm{\Delta }$). 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 (sequence A000594 in the OEIS):

$n$	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16
$\tau (n)$	1	−24	252	−1472	4830	−6048	−16744	84480	−113643	−115920	534612	−370944	−577738	401856	1217160	987136

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 $\tau (n)$:

• $\tau (mn)=\tau (m)\tau (n)$ if $\gcd (m,n)=1$ (meaning that $\tau (n)$ is a multiplicative function)
• $\tau (p^{r+1})=\tau (p)\tau (p^r)-p^{11}\tau (p^{r-1})$ for $p$ prime and $r>0$.
• $|\tau (p)|\le 2p^{11/2}$ for all primes $p$.

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 $k\in \mathbb{\mathbb{Z}}$ and $n\in \mathbb{\mathbb{N}}$, the Divisor function ${\sigma }_k(n)$ is 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 ${\sigma }_k(n)$. Here are some:

1. $\tau (n)\equiv {\sigma }_{11}(n)(\mathrm{mod}{2}^{11})\text{ for }n\equiv 1(\mathrm{mod}8)$
2. $\tau (n)\equiv 1217{\sigma }_{11}(n)(\mathrm{mod}{2}^{13})\text{ for }n\equiv 3(\mathrm{mod}8)$
3. $\tau (n)\equiv 1537{\sigma }_{11}(n)(\mathrm{mod}{2}^{12})\text{ for }n\equiv 5(\mathrm{mod}8)$
4. $\tau (n)\equiv 705{\sigma }_{11}(n)(\mathrm{mod}{2}^{14})\text{ for }n\equiv 7(\mathrm{mod}8)$
5. $\tau (n)\equiv n^{-610}{\sigma }_{1231}(n)(\mathrm{mod}{3}^6)\text{ for }n\equiv 1(\mathrm{mod}3)$
6. $\tau (n)\equiv n^{-610}{\sigma }_{1231}(n)(\mathrm{mod}{3}^7)\text{ for }n\equiv 2(\mathrm{mod}3)$
7. $\tau (n)\equiv n^{-30}{\sigma }_{71}(n)(\mathrm{mod}{5}^3)\text{ for }n\equiv ̸0(\mathrm{mod}5)$
8. $\tau (n)\equiv n{\sigma }_9(n)(\mathrm{mod}7)$
9. $\tau (n)\equiv n{\sigma }_9(n)(\mathrm{mod}{7}^2)\text{ for }n\equiv 3,5,6(\mathrm{mod}7)$
10. $\tau (n)\equiv {\sigma }_{11}(n)(\mathrm{mod}691).$

For $p\ne 23$ prime, we have

11. $\tau (p)\equiv 0(\mathrm{mod}23)\text{ if }\left(\frac{p}{23}\right)=-1$
12. $\tau (p)\equiv {\sigma }_{11}(p)(\mathrm{mod}23^2)\text{ if }p\text{ is of the form }{a}^2+23b^2$
13. $\tau (p)\equiv -1(\mathrm{mod}23)\text{ otherwise}.$

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 $\sigma (n)$ is the sum of the positive divisors of $n$.

Conjectures on the tau function

Suppose that $f$ is a weight-$k$ integer newform and the Fourier coefficients $a(n)$ are integers. Consider the problem:

Given that $f$ does not have complex multiplication, do almost all primes $p$ have the property that $a(p)\equiv ̸0(\mathrm{mod}p)$ ?

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 $a(n)(\mathrm{mod}p)$ for $n$ coprime to $p$, it is unclear how to compute $a(p)(\mathrm{mod}p)$. The only theorem in this regard is Elkies' famous result for modular elliptic curves, which guarantees that there are infinitely many primes $p$ such that $a(p)=0$, which thus are congruent to 0 modulo $p$. There are no known examples of non-CM $f$ with weight greater than 2 for which $a(p)\equiv ̸0(\mathrm{mod}p)$ for infinitely many primes $p$ (although it should be true for almost all $p$. There are also no known examples with $a(p)\equiv 0(\mathrm{mod}p)$ for infinitely many $p$. Some researchers had begun to doubt whether $a(p)\equiv 0(\mathrm{mod}p)$ for infinitely many $p$. As evidence, many provided Ramanujan's $\tau (p)$ (case of weight 12). The only solutions up to $10^{10}$ to the equation $\tau (p)\equiv 0(\mathrm{mod}p)$ are 2, 3, 5, 7, 2411, and (sequence A007659 in the OEIS).

conjectured that $\tau (n)\ne 0$ for all $n$, an assertion sometimes known as Lehmer's conjecture. Lehmer verified the conjecture for $n$ up to (Apostol 1997, p. 22). The following table summarizes progress on finding successively larger values of $N$ for which this condition holds for all $n\le N$.

$N$	reference
	Lehmer (1947)
	Lehmer (1949)
	Serre (1973, p. 98), Serre (1985)
	Jennings (1993)
	Jordan and Kelly (1999)
	Bosman (2007)
	Zeng and Yin (2013)
	Derickx, van Hoeij, and Zeng (2013)

Ramanujan's L-function

Ramanujan's $L$-function is defined by

$L(s)=\sum _{n\ge 1}\frac{\tau (n)}{n^s}$

if $\mathrm{R}\mathrm{e}(s)>6$ and by analytic continuation otherwise. It satisfies the functional equation

$\frac{L(s)\mathrm{\Gamma }(s)}{(2\pi {)}^s}=\frac{L(12-s)\mathrm{\Gamma }(12-s)}{(2\pi {)}^{12-s}},s\notin {\mathbb{\mathbb{Z}}}_0^{-},12-s\notin {\mathbb{\mathbb{Z}}}_0^{-}$

and has the Euler product

$L(s)=\prod _{p\text{prime}}\frac{1}{1-\tau (p)p^{-s}+p^{11-2s}},\mathrm{R}\mathrm{e}(s)>7.$

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