I'm currently working through Fundamentals of Number Theory by LeVeque, and one of the problems asks whether I can find some relationship between $τ(m)$, $τ(n)$ and $τ(mn)$, where $τ$ is the number of positive integers that divide its input.
The best I've been able to find is
$$τ(m)τ(n) = τ(mn) + R$$
where
$$R = τ\bigl(ε(m, n)\bigl)τ\bigl(ε(n, m)\bigl) \biggl[ τ\bigl(\frac{m}{ε(m, n)}\bigl) τ\bigl(\frac{n}{ε(n, m)}\bigl) - τ\bigl(\frac{mn}{ε(m, n)ε(n, m)}\bigl) \biggl]$$
where I've defined $ε(x, y)$ as a function which outputs $x$ after it is divided to the point of no longer sharing prime factors with $y$ (so, if $x=y$ then $ε(x, y)=1$, and if $gcd(x, y)=1$ then $ε(x, y)=x$, etc.).
I'm fairly sure about the logic behind this result and have tested it successfully. However, it isn't particularly useful or elegant.
I have two questions:
- What established relationships between $τ(m)$, $τ(n)$ and $τ(mn)$ are there in the literature?
- Does there exist a function which does what I've defined $ε(x, y)$ to do?