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Source. Croot, published paper, p. 233, Lemma 1 and equation (1). Croot attributes this to Lemma 3.1 of Canfield, Erdős and Pomerance, “On a problem of Oppenheim concerning ‘factorisatio numerorum’,” Journal of Number Theory 17 (1983), 1–28. That paper numbers no Lemma 3.1. Its Section 3 prints an unnumbered lemma (p. 9), Theorem 3.1 (p. 10, a lower bound only), and an unnumbered two-sided Corollary (p. 15), which the paper itself cites on p. 7 as the Corollary to Theorem 3.1. Lemma 1 follows from that Corollary with $u=c^{-1}\sqrt{\log x/\log\log x}$, for which $x^{1/u}=L(c,x)$. The canonical published source and its uniform smooth-number corollary are already filed separately. Their analytic proof remains external here.

Define, for $x>e$ and $y\ge2$,

$$T(x)=\sqrt{\log x\log\log x},\qquad L(c,x)=e^{cT(x)},$$

and

$$\psi(x,y)=\#\{1\le n\le x:p\mid n,\ p\text{ prime}\Longrightarrow p\le y\}.$$

Exact external input. For each fixed $c>0$,

$$\psi(x,L(c,x)) =x\exp\left(-\left(\frac1{2c}+o(1)\right)T(x)\right) \quad(x\longrightarrow\infty).$$

Equivalently, for every $\eta>0$, the count eventually lies between $x/L(1/(2c)+\eta,x)$ and $x/L(1/(2c)-\eta,x)$. No uniformity in an arbitrary varying $c$ is asserted. The lower-bound construction uses monotonicity between fixed nearby values of $c$ to handle its particular varying parameters.

Proof scope. This page records the precise analytic theorem used by Croot. The external Canfield–Erdős–Pomerance proof has not been reconstructed here. The prime-power variant following (1) is a separate, complete relative deduction.

Source corrections. Both displayed definitions of $\psi$ and $\psi^*$ on p. 233 print $n\le y$ where $n\le x$ is required. Taken literally, they would not depend on $x$ and could not satisfy (1). The arXiv v1 and the author manuscript date reference [1] to 1980; the published bibliography gives 1983.

Bears on. Theorem 1, the construction, and Problem 202.