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$\tau$ function (Definition)

The $ \tau$ function, also called the divisor function, takes a positive integer as its input and gives the number of positive divisors of its input as its output. For example, since $ 1$, $ 2$, and $ 4$ are all of the positive divisors of $ 4$, we have $ \tau (4)=3$. As another example, since $ 1$, $ 2$, $ 5$, and $ 10$ are all of the positive divisors of $ 10$, we have $ \tau (10)=4$.

The $ \tau$ function behaves according to the following two rules:

1. If $ p$ is a prime and $ k$ is a nonnegative integer, then $ \tau(p^k)=k+1$.

2. If $ \gcd(a,b)=1$, then $ \tau(ab)=\tau(a)\tau(b)$.

Because these two rules hold for the $ \tau$ function, it is a multiplicative function.

Note that these rules work for the previous two examples. Since $ 2$ is prime, we have $ \tau(4)=\tau(2^2)=2+1=3$. Since $ 2$ and $ 5$ are distinct primes, we have $ \tau(10)=\tau(2\cdot 5)=\tau(2)\tau(5)=(1+1)(1+1)=4$.

If $ n$ is a positive integer, the number of prime factors of $ x^n-1$ over $ \mathbb{Q}[x]$ is $ \tau(n)$. For example, $ x^9-1=(x^3-1)(x^6+x^3+1)=(x-1)(x^2+x+1)(x^6+x^3+1)$ and $ \tau(9)=3$.

The $ \tau$ function is extremely useful for studying cyclic rings.

The sequence $ \{\tau(n)\}$ appears in the OEIS as sequence A000005.



"$\tau$ function" is owned by Wkbj79.
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See Also: divisor, Dirichlet hyperbola method, $2^{\omega(n)} \le \tau(n) \le 2^{\Omega(n)}$, divisibility, values of $n$ for which $\varphi(n)=\tau(n)$

Other names:  divisor function

Attachments:
proof that $\tau(n)$ is the number of positive divisors of $n$ (Proof) by Wkbj79
the divisor function is multiplicative (Theorem) by yark
$\displaystyle \sum_{n \le x} (\tau(n))^a=O_a(x(\log x)^{2^a-1})$ for $a \ge 0$ (Theorem) by Wkbj79
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Cross-references: OEIS, sequence, cyclic rings, multiplicative function, prime, divisors, number, integer, positive
There are 23 references to this entry.

This is version 18 of $\tau$ function, born on 2003-03-10, modified 2008-05-16.
Object id is 4085, canonical name is TauFunction.
Accessed 4331 times total.

Classification:
AMS MSC: 11A25 (Number theory :: Elementary number theory :: Arithmetic functions; related numbers; inversion formulas)
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