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For every rational $q>0$,

$$\sum_{\varphi(d)/d=q}\frac1d\le1.$$

The complete proof is the support computation in Lemma 2.1. Its exact equality cases and the strict half-gap are recorded in Remarks 2.2–2.3.

The introductory motivation on published p.796 reverses a subtraction. If $n_1=d_1p_1<n_2=d_2p_2$ have the same ratio $q$ and spacing at least $D_0=\max\mathcal D$, the correct formula is

$$\varphi(n_2)-\varphi(n_1) =q(n_2-n_1+d_1-d_2)\ge q(D_0+d_1-d_2)\ge qd_1>0.$$

The source's further counterfactual prime-sieve counting argument is motivational only and is not claimed as a separately reconstructed proof here. It is unnecessary for Lemma 2.1 or the main theorem.

Source. Tao, published paper, published pp.796–797, Proposition 1.4 and its motivation. This page uses that published version.

Bears on. Problem 49.