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Claim. For a set $\mathcal P$ of primes let $S(x,\mathcal P)$ count the integers $n\le x$ divisible by no prime of $\mathcal P$, and let $G(x,K)$ be the least value of $S(x,\mathcal P)/x$ over sets $\mathcal P$ with $\sum_{p\in\mathcal P}1/p\le K$. Corollary 1 of A. Hildebrand, Quantitative mean value theorems for nonnegative multiplicative functions. II, Acta Arith. 48 (1987), 209--260, states that uniformly for $x\ge2$ and $0<K\le c_1\log\log x$,

$$G(x,K)=\rho(e^K)\Bigl(1+O\bigl((\log x)^{-c_2}\bigr)\Bigr),$$

with absolute positive constants $c_1,c_2$ and $\rho$ the Dickman function. For the problem this says: if $A\subseteq\{2,\ldots,N\}$ consists of primes with $\sum_{p\in A}1/p\le C$, then the number of $m\le N$ divisible by no element of $A$ is at least $\rho(e^C)N(1+O((\log N)^{-c_2}))$, and the primes between $N^{e^{-C}}$ and $N$ attain $\rho(e^C)N$ asymptotically, the form of the Erdős-Ruzsa conjecture as the introduction states it. The power-of-$\log$ error is stated for the minimum $G$ only; the corollary does not state an error for the prime tail (the proof's upper estimate for $G$, Section 8, is computed from that tail). The paper derives the corollary from its Theorem 2, the sharp lower bound for the mean value of a nonnegative multiplicative function, applied to the indicator of the integers free of primes in $\mathcal P$; the source card hildebrand_1987_quantitative_mean_value_theorems_nonnegative_multiplicative transcribes Theorem 2 and the example showing its Dickman factor is best possible. The paper presents the corollary as a quantitative form of the conjecture of Erdős and Ruzsa (Problem 1, p. 386, of On the small sieve, I; Hildebrand's introduction cites it as Problem 2 of that paper, whose Problem 2 is the residue-class question, and the locator here follows the paper itself) that the minimum defining $G(x,K)$ is asymptotically attained by the primes between $x^{e^{-K}}$ and $x$; Erdős and Ruzsa had proved the weaker $G(x,K)\ge e^{-e^{cK}}$ in the normalization used here (their Theorem 1, where $G$ counts integers and the bound carries a factor $x$ that display (1.4) omits).

Covers. The case in which $A$ consists of primes: the least number of $m\le N$ divisible by no element of such an $A$ is $(\rho(e^C)+o(1))N$, with the error term a power of $\log N$. The claim says nothing about sets with composite elements. Erdős and Ruzsa had asserted the extension to all pairwise coprime $A$ relative to the prime case without writing out its proof (their display (1.12); their claim page). That display and this corollary together would give the corrected Statement, and Tao's claim page deduces the extension from this corollary with a written proof. The exact minimizer for a given $N$, the site's wording, is determined by none of them.

Acceptance. Refereed: Acta Arithmetica, volume 48, issue 3, pages 209--260, DOI 10.4064/aa-48-3-209-260, published under the journal's Creative Commons Attribution license. Reviewed: the site's curator, Thomas Bloom, who is independent of the author, writes in the commentary (page last edited 28 May 2026, accessed 2026-09-05) that Hildebrand proved the weak form of the conjecture when $A$ is a set of primes, answering the question of Erdős and Ruzsa; a thread comment of 4 February 2026 first pointed the thread to the corollary.

Date. The paper appeared in 1987; the issue month is not recorded here, so the page name uses the first day of that year.

Depends on. No page of this wiki.