Any way to simplify this gcd totient function

Question

I have the following expression

$$\frac{gcd(a,b)}{\varphi(gcd(a,b))}$$

$a,b$ are known positive integers. Is there any way to rephrase this or simplify it?

Answer 1

In general, $$\frac{n}{\phi(n)} = \prod_{p|n} \frac{p}{p-1}$$

where the product is over all primes that divide $n$. So in your case, with $n=(a,b)$ this becomes:

$$\prod_{p|a \land p|b} \frac{p}{p-1}$$

In don't think there is any better formula. Since $\gcd(a,b)$ can be any natural number, there is no real trick.