Ratio of totient $\varphi(n)/\varphi(n+1)$

Question

Is there a general formula for the ratio of totients of two consecutive integers? It occurred to me in a different context, where I was trying to find a different way of estimating the probability of drawing a random rational number in the interval $(0,1)$. One could represent a rational number as an ordered pair of co-primes $(m,n)$ with $m<n$. Suppose we put a uniform, finitely additive measure on the set $\{(m,n)\in\mathbb{N}_{>0}: m, n \text{co-prime}, m<n\}$. Then fixing $n$, there are only $\varphi(n)$ many rational numbers with denominator $n$, which is what brought me to the question of $\varphi(n)/\varphi(n+1)$. Ideally it would be great if the formula could be a function of $n$ alone (sorry this might still be vague). Many thanks in advance for any help.

Answer 1

If $\;n=p_1^{a_1}\cdot\ldots\cdot p_m^{a_m}\;,\;\;p_i,\,a_i\in\Bbb N\;,\;\;p_i$ primes, then we know that

$$\phi(n)=n\prod_{k=1}^m\left(1-\frac1{p_k}\right)$$

so you can then write

$$\frac{\phi(n)}{\phi(n+1)}=\frac n{n+1}\prod_{p\mid n,\,q\mid (n+1)}\left(1-\frac1p\right)\left(1-\frac1q\right)^{-1}$$

It doesn't really look very nice a formula, though...