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The mass in Lemma 2.1 equals one exactly when $q=1$ or $q=1/2$. Every other $q>0$ has mass at most $1/2$. Also,

$$\frac{\varphi(m)}m=\frac{\varphi(n)}n \quad\Longleftrightarrow\quad \{p:p\mid m\}=\{p:p\mid n\}.$$

Proof. In a nonempty fiber, the product $\prod_{p\in P}(p-1)^{-1}$ equals one precisely when every support prime is two. Thus $P=\varnothing$ or $P=\{2\}$, giving respectively $q=1$ and $q=1/2$. Any other support contains a prime at least three, whose factor is at most one-half; all remaining factors are at most one. Empty fibers have mass zero. Finally equal ratios have the same support by Lemma 2.1, and equal supports have the same ratio by the Euler product. $\square$

Source. Tao, published paper, published p.800, Remarks 2.2–2.3. This page uses that published version.

Bears on. Problem 49.