I am reading Sierpinski's book: Elementary theory of number, 1988.
In that book has the following problem (page 254):
Problem. Prove that for arbitrary natural numbers $a,b$ there exist infinitely many pairs of natural numbers $x,y$ such that $\displaystyle \frac{\varphi (x)}{\varphi (y)}=\frac{a}{b}.$
Solution in the book. Without loss of ganerality we may assume that $\gcd (a,b)=1$. Let $x=a^2b(kab+1),y=ab^2(kab+1)$, here $k$ is a positive integer, then $\displaystyle \frac{\varphi (x)}{\varphi (y)}=\frac{a}{b}.$
This is my question.
Conjecture. For arbitrary natural numbers $a,b$ there exist infinitely many pairs of natural numbers $x,y$ such that $\gcd (x,y)=1$ and $\displaystyle \frac{\varphi (x)}{\varphi (y)}=\frac{a}{b}.$
Can you help me?