Ramanujan tau function

The Ramanujan tau function is the arithmetic function $\tau\colon\mathbb{N}\to\mathbb{Z}$ such that, for all $q\in\mathbb{C}$ with $|q|<1$ ,$$ q\prod_{k=1}^{\infty}(1-q^k)^{24}=\sum_{n=1}^{\infty} \tau(n)q^n.$$ 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 $q$ , $q^2$ , and $q^3$ , respectively, of the expression$$ q\prod_{k=1}^{\infty}(1-q^k)^{24}.$$ Note that we only need to consider $k=1$ and $k=2$ , since higher values of $k$ yield powers of $q$ that are too large. Thus:

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

Although the values of $|\tau(n)|$ seem to increase rapidly as $n$ 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 $p$ and any $n\in\mathbb{N}$ ,$$ \tau(p^{n+1})=\tau(p)\tau(p^n)-p^{11}\tau(p^{n-1}).$$
• For any prime $p$ ,$$ |\tau(p)|\le 2p^{\frac{11}{2}}.$$

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

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

Bibliography

1

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