Ramanujan tau function

Values of Template:Math for Template:Math with a logarithmic scale. The blue line picks only the values of Template:Mvar that are multiples of 121.

The Ramanujan tau function, studied by Template:Harvs, is the function $\tau :\mathbb{N}\to \mathbb{Z}$ 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 Template:Math with Template:Math, $\varphi$ is the Euler function, Template:Mvar is the Dedekind eta function, and the function Template:Math 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 Template:Harvtxt.

Values

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

Template:Mvar	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16
Template: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 conjectures

Template:Harvtxt observed, but did not prove, the following three properties of Template:Math:

• Template:Math if Template:Math (meaning that Template:Math is a multiplicative function)
• Template:Math for Template:Mvar prime and Template:Math.
• Template:Math for all primes Template:Mvar.

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 function

For Template:Math and Template:Math, the Divisor function Template:Math is the sum of the Template:Mvarth powers of the divisors of Template:Mvar. The tau function satisfies several congruence relations; many of them can be expressed in terms of Template:Math. Here are some:

1. $\tau (n)\equiv {\sigma }_{11}(n)mod{2}^{11}\text{ for }n\equiv 1mod8$
2. $\tau (n)\equiv 1217{\sigma }_{11}(n)mod{2}^{13}\text{ for }n\equiv 3mod8$
3. $\tau (n)\equiv 1537{\sigma }_{11}(n)mod{2}^{12}\text{ for }n\equiv 5mod8$
4. $\tau (n)\equiv 705{\sigma }_{11}(n)mod{2}^{14}\text{ for }n\equiv 7mod8$
5. $\tau (n)\equiv n^{-610}{\sigma }_{1231}(n)mod{3}^6\text{ for }n\equiv 1mod3$
6. $\tau (n)\equiv n^{-610}{\sigma }_{1231}(n)mod{3}^7\text{ for }n\equiv 2mod3$
7. $\tau (n)\equiv n^{-30}{\sigma }_{71}(n)mod{5}^3\text{ for }n\equiv ̸0mod5$
8. $\tau (n)\equiv n{\sigma }_9(n)mod7$
9. $\tau (n)\equiv n{\sigma }_9(n)mod{7}^2\text{ for }n\equiv 3,5,6mod7$
10. $\tau (n)\equiv {\sigma }_{11}(n)mod691.$

For Template:Math prime, we have

11. $\tau (p)\equiv 0mod23\text{ if }\left(\frac{p}{23}\right)=-1$
12. $\tau (p)\equiv {\sigma }_{11}(p)mod23^2\text{ if }p\text{ is of the form }{a}^2+23b^2$
13. $\tau (p)\equiv -1mod23\text{ otherwise}.$

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 Template:Math is the sum of the positive divisors of Template:Mvar.

Conjectures on τ(n)

Suppose that Template:Mvar is a weight-Template:Mvar integer newform and the Fourier coefficients Template:Math are integers. Consider the problem:

Given that Template:Mvar does not have complex multiplication, do almost all primes Template:Mvar have the property that Template: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 Template:Math for Template:Mvar coprime to Template:Mvar, it is unclear how to compute Template:Math. The only theorem in this regard is Elkies' famous result for modular elliptic curves, which guarantees that there are infinitely many primes Template:Mvar such that Template:Math, which thus are congruent to 0 modulo Template:Math. There are no known examples of non-CM Template:Mvar with weight greater than 2 for which Template:Math for infinitely many primes Template:Mvar (although it should be true for almost all Template:Mvar). There are also no known examples with Template:Math for infinitely many Template:Mvar. Some researchers had begun to doubt whether Template:Math for infinitely many Template:Mvar. As evidence, many provided Ramanujan's Template:Math (case of weight 12). The only solutions up to 10¹⁰ to the equation Template:Math are 2, 3, 5, 7, 2411, and Template:Val Template:OEIS.

Template:Harvtxt conjectured that Template:Math for all Template:Mvar, an assertion sometimes known as Lehmer's conjecture. Lehmer verified the conjecture for Template:Mvar up to Template:Val (Apostol 1997, p. 22). The following table summarizes progress on finding successively larger values of Template:Mvar for which this condition holds for all Template:Math.

Template:Mvar	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-function

Ramanujan's L-function is defined by

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

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

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

and has the Euler product

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

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

References

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