Ramanujan tau function

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

</math> 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 Template:Harvs, 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>q=\exp(2\pi iz)</math> with <math>\mathrm{Im}(z)>0</math>, <math>\phi</math> is the Euler function, <math>\eta</math> is the Dedekind eta function, and the function <math>\Delta(z)</math> is a holomorphic cusp form of weight 12 and level 1, known as the discriminant modular form (some authors, notably Apostol, write <math>\Delta/(2\pi)^{12}</math> instead of <math>\Delta</math>). 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 Template:Harvtxt.

ValuesEdit

The first few values of the tau function are given in the following table (sequence A000594 in the OEIS):

<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.<ref>Template:Cite OEIS</ref>

Ramanujan's conjecturesEdit

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

<math>\tau(mn)=\tau(m)\tau(n)</math> if <math>\gcd(m,n)=1</math> (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>.
<math>|\tau(p)|\leq 2p^{11/2}</math> for all primes <math>p</math>.

The first two properties were proved by Template:Harvtxt 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 functionEdit

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:<ref name=swd>Page 4 of Template:Harvnb</ref>

<math>\tau(n)\equiv\sigma_{11}(n)\ \bmod\ 2^{11}\text{ for }n\equiv 1\ \bmod\ 8</math><ref name=kolberg>Due to Template:Harvnb</ref>
<math>\tau(n)\equiv 1217 \sigma_{11}(n)\ \bmod\ 2^{13}\text{ for } n\equiv 3\ \bmod\ 8</math><ref name=kolberg/>
<math>\tau(n)\equiv 1537 \sigma_{11}(n)\ \bmod\ 2^{12}\text{ for }n\equiv 5\ \bmod\ 8</math><ref name=kolberg/>
<math>\tau(n)\equiv 705 \sigma_{11}(n)\ \bmod\ 2^{14}\text{ for }n\equiv 7\ \bmod\ 8</math><ref name=kolberg/>
<math>\tau(n)\equiv n^{-610}\sigma_{1231}(n)\ \bmod\ 3^{6}\text{ for }n\equiv 1\ \bmod\ 3</math><ref name=ashworth>Due to Template:Harvnb</ref>
<math>\tau(n)\equiv n^{-610}\sigma_{1231}(n)\ \bmod\ 3^{7}\text{ for }n\equiv 2\ \bmod\ 3</math><ref name=ashworth/>
<math>\tau(n)\equiv n^{-30}\sigma_{71}(n)\ \bmod\ 5^{3}\text{ for }n\not\equiv 0\ \bmod\ 5</math><ref>Due to Lahivi</ref>
<math>\tau(n)\equiv n\sigma_{9}(n)\ \bmod\ 7</math><ref name=Lehmer>Due to D. H. Lehmer</ref>
<math>\tau(n)\equiv n\sigma_{9}(n)\ \bmod\ 7^2\text{ for }n\equiv 3,5,6\ \bmod\ 7</math><ref name=Lehmer/>
<math>\tau(n)\equiv\sigma_{11}(n)\ \bmod\ 691.</math><ref>Due to Template:Harvnb</ref>

For <math>p\neq 23</math> prime, we have<ref name=swd/><ref>Due to Template:Harvnb</ref>

<math>\tau(p)\equiv 0\ \bmod\ 23\text{ if }\left(\frac{p}{23}\right)=-1</math>
<math>\tau(p)\equiv \sigma_{11}(p)\ \bmod\ 23^2\text{ if } p\text{ is of the form } a^2+23b^2</math><ref>Due to J.-P. Serre 1968, Section 4.5</ref>
<math>\tau(p)\equiv -1\ \bmod\ 23\text{ otherwise}.</math>

Explicit formulaEdit

In 1975 Douglas Niebur proved an explicit formula for the Ramanujan tau function:<ref>Template:Cite journal</ref>

<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 the positive divisors of <math>n</math>.

Conjectures on the tau functionEdit

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 Template:Val (sequence A007659 in the OEIS).<ref name=Lygeros>Template:Cite journal</ref>

Template:Harvtxt 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 Template:Val (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>.

<math>N</math>	reference
Template:Val	Lehmer (1947)
Template:Val	Lehmer (1949)
Template:Val	Serre (1973, p. 98), Serre (1985)
Template:Val	Jennings (1993)
Template:Val	Jordan and Kelly (1999)
Template:Val	Bosman (2007)
Template:Val	Zeng and Yin (2013)
Template:Val	Derickx, van Hoeij, and Zeng (2013)

Ramanujan's L-functionEdit

Ramanujan's <math>L</math>-function is defined by

if <math>\mathrm{Re}(s)>6</math> and by analytic continuation otherwise. It satisfies 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>

and has 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>.

NotesEdit

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