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Source: arXiv v2, p. 6, Lemma 3.4. The source gives it as a consequence of Theorem 16.3 in Dimitris Koukoulopoulos, The Distribution of Prime Numbers, Graduate Studies in Mathematics 203, American Mathematical Society, 2019.

Exact external statement

Let $x\ge y\ge2$, assume

$$y\ge(\log x)^3,$$

and set $u=\log x/\log y$. Then

$$\sum_{\substack{d>y^u\\P^+(d)\le y}}\frac1d \ll\frac{\log y}{u^u}, \tag{1}$$

with an absolute implied constant. Since $y^u=x$, the sum is the reciprocal tail over $y$-smooth integers greater than $x$.

The book theorem proving (1) is an external analytic input. This page records its exact hypothesis and conclusion rather than reconstructing that proof.