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Statement

Theorem 1 (p. 251). For an infinite increasing sequence of integers $b_1<b_2<\cdots$, with $g(n)$ "the number of solutions of $n=b_ib_j$": if $g(n)>0$ for every $n>n_0$ (display (1)), then $\limsup_{n\to\infty}g(n)=\infty$ (display (2)).

The page does not say whether $i=j$ is allowed or whether ordered pairs are counted. The paper's abstract (p. 251) states the stronger form "If $g(n)>0$ for a sequence $n$ of positive upper density then $\limsup g(n)=\infty$", which Theorem 4 (p. 252) implies.

Source. P. Erdős, On the multiplicative representation of integers, Israel J. Math. 2 (1964), no. 4, 251--261 (received 17 December 1964); Theorem 1 on printed p. 251 (PDF p. 1 of the eleven-page scan), read on the page image. The proof is completed on p. 254 together with Theorem 2 ("which proves Theorems 1 and 2").

Read depth. Claims checked: the statement and the two sentences that follow it (Raikov's theorem (3) and the reduction) were read clause by clause on the page image. The proof (pp. 252--254, through Theorem 2 and the Lemma) was read on the page images for its structure and not checked step by step.

Proof pointer

Page 251: by Raikov's theorem (the paper's [5]), hypothesis (1) gives $B(x)=\sum_{b_i\le x}1>c_1x/(\log x)^{1/2}$ for infinitely many $x$ (display (3)), so it is enough to derive (2) from (3) holding for infinitely many $x$. Page 252 observes that Theorem 1 would follow from $u_l(n)=O(n/(\log n)^{1/2})$, and Theorem 2 supplies the smaller bound $u_{2^k}(n)<c_2n(\log\log n)^{k+1}/\log n$, proved on pp. 253--254 from the paper's Lemma and Landau's asymptotic for integers with $k$ prime factors.

Dependencies

Raikov's theorem (the paper's [5]) and Theorem 2.

Bears on

• Problem 796: the paper's qualitative result on representation functions; the problem's quantitative question concerns Theorem 3 of the same paper.