PlanetMath: Ramanujan tau function

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Ramanujan tau function

(Definition)

The Ramanujan tau function is the arithmetic function $\tau\colon\mathbb{N}\to\mathbb{Z}$ such that, for all $q\in\mathbb{C}$ with $\vert q\vert<1$ ,

$\displaystyle q\prod_{k=1}^{\infty}(1-q^k)^{24}=\sum_{n=1}^{\infty} \tau(n)q^n.$

Thus, the Ramanujan tau function is the generating function for the Weierstrass $\Delta$ function.

Determining values of the Ramanujan tau function directly can be somewhat involved. For example, the values of $\tau(1)$ , $\tau(2)$ , and $\tau(3)$ will be determined:

To determine $\tau(1)$ , $\tau(2)$ , and $\tau(3)$ , we need to find the coefficient of , , and , respectively, of the expression

$\displaystyle q\prod_{k=1}^{\infty}(1-q^k)^{24}.$

Note that we only need to consider

and

, since higher values of

yield powers of

that are too large. Thus:

$\displaystyle q(1-q)^{24}(1-q^2)^{24}$	$\displaystyle =q(1-24q+276q^2-\dots)(1-24q^2+\dots)$
	$\displaystyle =q(1-24q+276q^2-\dots-24q^2+576q^3-\dots)$
	$\displaystyle =q(1-24q+252q^2-\dots)$
	$\displaystyle =q-24q^2+252q^3-\dots$

Hence, $\tau(1)=1$ , $\tau(2)=-24$ , and $\tau(3)=252$ .

The sequence $\{\tau(n)\}$ appears in the OEIS as sequence A000594.

Although the values of $\vert\tau(n)\vert$ seem to increase rapidly as increases, the conjecture that $\tau(n)\neq 0$ for all $n\in\mathbb{N}$ has not yet been proven. This conjecture is known as Lehmer's conjecture.

The Ramanujan tau function has the following properties:

It is a multiplicative function: For $a,b\in\mathbb{N}$ with $\gcd(a,b)=1$ , we have $\tau(ab)=\tau(a)\tau(b)$ .
For any prime and any $n\in\mathbb{N}$ ,
$\displaystyle \tau(p^{n+1})=\tau(p)\tau(p^n)-p^{11}\tau(p^{n-1}).$
For any prime ,
$\displaystyle \vert\tau(p)\vert\le 2p^{\frac{11}{2}}.$

Ramanujan asserted that $\tau$ satisfies several congruences, all of which have been proven. Some simpler examples of such congruences include:

For any $n\in\mathbb{N}$ ,
$\displaystyle \tau(5n)\equiv 0\pmod 5.$
For any $n\in\mathbb{N}$ and for any nonnegative integer which is a quadratic residue modulo ,
$\displaystyle \tau(7n-r)\equiv 0\pmod 7.$
For any $n\in\mathbb{N}$ and for any nonnegative integer which is a quadratic residue modulo ,
$\displaystyle \tau(23n-r)\equiv 0\pmod{23}.$

Bibliography

1: Berndt, Bruce C. Number Theory in the Spirit of Ramanujan. Providence, RI: American Mathematical Society, 2006.

"Ramanujan tau function" is owned by Wkbj79.

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Other names:

Ramanujan's tau function

Also defines:

Lehmer's conjecture

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Cross-references: quadratic residue, integer, congruences, Ramanujan, prime, multiplicative function, properties, conjecture, OEIS, sequence, expression, coefficient, generating function, arithmetic function
There are 3 references to this entry.

This is version 9 of Ramanujan tau function, born on 2008-02-26, modified 2008-02-26.
Object id is 10333, canonical name is RamanujanTauFunction.
Accessed 588 times total.

Classification:

AMS MSC:	11F11 (Number theory :: Discontinuous groups and automorphic forms :: Modular forms, one variable)
	11A25 (Number theory :: Elementary number theory :: Arithmetic functions; related numbers; inversion formulas)

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