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Source. Theorem 1 and equation (3), printed p. 85; proof pp. 86–89 (PDF pp. 1–5).

Statement. There is an absolute constant $c>0$ such that, for every $\epsilon>0$, all sufficiently large real $x$ satisfy

$$\frac{x}{\exp((\log x)^{1/2+\epsilon})} < f(x)<\frac{x}{(\log x)^c}. \tag{1}$$

The function $f(x)$ is the maximum size of a disjoint progression family with distinct proper moduli at most $x$, as specified on the definitions page. In particular $f(x)=o(x)$.

Complete deduction from the original chains

The prime-chain construction gives the lower inequality in (1) for every fixed $\epsilon>0$ and all $x$ above an $\epsilon$-dependent threshold. Its residues and square-free count are both supplied in full.

For the upper inequality, Lemma 1 gives $f(x)\le F(x)$, and the complete upper proof of Theorem 2 gives $F(x)<x/(\log x)^c$ for an absolute $c>0$ and all sufficiently large $x$. That argument uses Lemmas 2–4 and the explicitly proved normal-order estimate; every essential same-paper deduction is linked there. Taking the larger of the two thresholds proves (1).

Finally $0\le f(x)/x<(\log x)^{-c}\to0$. This proves the Erdős–Stein density-zero conjecture addressed by the 1968 paper. The lower construction for the auxiliary $F$ is a separate limitation result and is not needed for this conclusion.

Scope. These are historical quantitative bounds. They do not identify the later sharp order or establish a current status assessment by themselves. Classical prime estimates, Bertrand's postulate and CRT are the exact external inputs identified separately; no fresh formal verification is claimed.

Bears on. Problem 202. The paper's disjoint systems and the accompanying reciprocal-sum bound also belong to the historical background of Problem 1190; its later tail optimization is not asserted or proved here.