Tau Function of Square-Free Integer is Power of 2

Theorem

Let $n$ be a square-free integer.

Let $\tau: \Z \to \Z$ be the tau function.

Then $\tau \left({n}\right) = 2^r$ for some $r \ge 1$.

The converse is not true in general.

That is, if $\tau \left({n}\right) = 2^r$ for some $r \ge 1$, it is not necessarily the case that $n$ is square-free.

Proof

Let $n$ be an integer such that $n \ge 2$, with prime decomposition $n = p_1^{k_1} p_2^{k_2} \ldots p_r^{k_r}$.

Then from Tau Function - Number of Positive Divisors we have that:

$\displaystyle \tau \left({n}\right) = \prod_{i=1}^r \left({k_i + 1}\right)$

• Let $n$ be square-free.

Then by definition $\forall i: 1 \le i \le r: k_i = 1$.

So:

$\displaystyle \tau \left({n}\right) = \prod_{i=1}^r \left({1 + 1}\right) = 2^r$

• Now let $n = p^3$ where $p$ is prime.

Then $n$ is not square-free as $p^2 \backslash n$.

However, $\tau \left({n}\right) = 3 + 1 = 2^2$.

$\blacksquare$