Ramanujan tau function

for <math>n<16,000</math> with a logarithmic scale. The blue line picks only the values of <math>n</math> that are multiples of 121.]]

The Ramanujan tau function, studied by , is the function <math>\tau : \mathbb{N}\to\mathbb{Z}</math> defined by the following identity:

:<math>\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),</math>

where <math display="inline">q=e^{2\pi iz}</math> with <math display="inline">\mathrm{Im}(z)>0</math>, <math display="inline">\phi</math> is the Euler function, <math display="inline">\eta</math> is the Dedekind eta function, and the function <math display="inline">\Delta(z)</math> is the modular discriminant.

Values

The first few values of the tau function are given in the following table :

{| class="wikitable" style="text-align:center"

! <math>n</math>

|1||2||3||4||5||6||7||8||9||10||11||12||13||14||15||16

! <math>\tau(n)</math>

|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 L-function

Because the modular discriminant <math display="inline">\Delta(z)</math> is a holomorphic cusp form of weight 12, Ramanujan's <math>L</math>-function, defined by

:<math>L(s)=\sum_{n\ge 1}\frac{\tau (n)}{n^s}</math>,

for <math>\mathrm{Re}(s)>7</math>, can be analytically continued for all complex numbers <math>s</math> via the functional equation

:<math>\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^{-},</math>

thus making <math display="inline">L(s)</math> an entire function. Ramanujan's <math>L</math>-function satisfy the Euler product

:<math>L(s)=\prod_{p\,\text{prime\frac{1}{1-\tau (p)p^{-s}+p^{11-2s,\quad \mathrm{Re}(s)>7.</math>

Ramanujan conjectured that all nontrivial zeros of <math>L</math> have real part equal to <math>6</math>.

Main properties

observed, but did not prove, the following two properties of <math>\tau(n)</math>:

<math>\tau(mn)=\tau(m)\tau(n)</math> if <math>m</math> and <math>n</math> are coprime (meaning that <math>\tau(n)</math> is a multiplicative function)
<math>\tau(p^{r+1})=\tau(p)\tau(p^r)-p^{11}\tau(p^{r-1})</math> for <math>p</math> prime and <math>r>0</math>.

The two properties are equivalent to both the Euler product for Ramanujan's <math>L</math>-function and the Hecke relation<math display="block"> \mathop\tau(m) \mathop\tau(n) = \sum_{d|(m,n)}d^{11}\mathop\tau\left(\frac{mn}{d^2}\right),\quad m,n\geq 1. </math>They were proved by . Ramanujan also conjectured the third property<math display="block"> |\tau(p)| \le 2 p^{\frac{11}2} </math>for primes <math>p</math>, which is called the Ramanujan conjecture. Assuming the first properties, Ramanujan noted that his conjecture is equivalent to the inequality <math display="inline"> |\tau(n)| \le d(n) n^{\frac{11}2} </math>, for all <math>n \ge 1</math>, where <math>d(n)</math> is the number-of-divisor function. The latter bound implies that both the Ramanujan <math>L</math>-function and its Euler product converge for <math>\mathrm{Re}(s)>\frac{13}2</math>. This 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 <math>k\in\mathbb{Z}</math> and <math>n\in\mathbb{N}</math>, the divisor function <math>\sigma_k(n)</math> is the sum of the <math>k</math>th powers of the divisors of <math>n</math>. The tau function satisfies several congruence relations; many of them can be expressed in terms of <math>\sigma_k(n)</math>. Here are some:

<math>\tau(n)\equiv\sigma_{11}(n) \pmod{2^{11\text{ for }n\equiv 1 \pmod{8}</math>
<math>\tau(n)\equiv 1217 \sigma_{11}(n) \pmod{2^{13\text{ for } n\equiv 3 \pmod{8}</math>
<math>\tau(n)\equiv n^{-610}\sigma_{1231}(n) \pmod{3^{7\text{ for }n\equiv 2 \pmod{3}</math>
<math>\tau(n)\equiv n\sigma_{9}(n) \pmod{7}</math>
<math>\tau(n)\equiv n\sigma_{9}(n) \pmod{7^2}\text{ for }n\equiv 3,5,6 \pmod{7}</math>

For <math>p\neq 23</math> prime, we have

<li><math>\tau(p)\equiv 0 \pmod{23}\text{ if }\left(\frac{p}{23}\right)=-1</math>

<li><math>\tau(p)\equiv \sigma_{11}(p) \pmod{23^2}\text{ if } p\text{ is of the form } a^2+23 b^2</math>

<li><math>\tau(p)\equiv -1 \pmod{23}\text{ otherwise}.</math>

</ol>

Explicit formulas

In 1972, Ian G. Macdonald proved an explicit formula for the Ramanujan tau function<math display="block">\tau(n)

\frac{1}{1!\,2!\,3!\,4!}

\sum_{

\begin{smallmatrix}

x_1+x_2+x_3+x_4+x_5=0\\

x_1^2+x_2^2+x_3^2+x_4^2+x_5^2=10n\\

x_i\equiv i \pmod 5,\; i=1,\dots,5

\end{smallmatrix}

}

\prod_{1\le i<j\le 5}(x_i-x_j).</math>In 1975 Douglas Niebur proved the formula

:<math>\tau(n)=n^4\sigma(n)-24\sum_{i=1}^{n-1}i^2(35i^2-52in+18n^2)\sigma(i)\sigma(n-i),</math>

where <math>\sigma(n)</math> is the sum-of-divisor function.

Conjectures on the tau function

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

: Given that <math>f</math> does not have complex multiplication, do almost all primes <math>p</math> have the property that <math>a(p)\not\equiv 0\pmod{p}</math> ?

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 <math>a(n)\pmod{p}</math> for <math>n</math> coprime to <math>p</math>, it is unclear how to compute <math>a(p)\pmod{p}</math>. The only theorem in this regard is Elkies' famous result for modular elliptic curves, which guarantees that there are infinitely many primes <math>p</math> such that <math>a(p)=0</math>, which thus are congruent to 0 modulo <math>p</math>. There are no known examples of non-CM <math>f</math> with weight greater than 2 for which <math>a(p)\not\equiv 0\pmod{p}</math> for infinitely many primes <math>p</math> (although it should be true for almost all <math>p</math>. There are also no known examples with <math>a(p)\equiv 0 \pmod{p}</math> for infinitely many <math>p</math>. Some researchers had begun to doubt whether <math>a(p)\equiv 0 \pmod{p}</math> for infinitely many <math>p</math>. As evidence, many provided Ramanujan's <math>\tau(p)</math> (case of weight 12). The only solutions up to <math>10^{10}</math> to the equation <math>\tau(p)\equiv 0\pmod{p}</math> are 2, 3, 5, 7, 2411, and .

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

{| class="wikitable"

! <math>N</math> !! reference

| align="right" | || Lehmer (1947)

| align="right" | || Lehmer (1949)

| align="right" | || Serre (1973, p. 98), Serre (1985)

| align="right" | || Jennings (1993)

| align="right" | || Jordan and Kelly (1999)

| align="right" | || Bosman (2007)

| align="right" | || Zeng and Yin (2013)

| align="right" | || Derickx, van Hoeij, and Zeng (2013)