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Statement

Theorem 4 (p. 252, quoted). "To every $c$ and $l$ there is an $n_0=n_0(c,l)$ so that if $n>n_0$ and $b_1<\dots<b_s\le n$ is such that the number $N(n)$ of integers $t<n$ which can be written in the form $b_ib_j$ is greater than $c\,n$ then there is an $m$ with $g(m)>l$."

Here $g(m)$ is the number of solutions of $m=b_ib_j$ (p. 251); the page does not say whether $i=j$ is allowed. The paper notes that Theorem 4 implies Theorem 1 but not Theorems 2 and 3 (p. 252), and its abstract (p. 251) states the consequence that $g(n)>0$ on a set of positive upper density forces $\limsup g(n)=\infty$.

Source. P. Erdős, On the multiplicative representation of integers, Israel J. Math. 2 (1964), no. 4, 251--261; Theorem 4 on printed p. 252, proof on pp. 254--255.

Read depth. Claims checked: the statement was read clause by clause on the page image. The proof (pp. 254--255) was read for its structure and not checked step by step.

Proof pointer

Pages 254--255. With $B(x)$ the number of $b_i\le x$, the paper shows that there is $\varepsilon=\varepsilon(c)>0$ such that for every $T$ there is $n_0(T,\varepsilon)$ for which $n>n_0$ and $N(n)>cn$ give some $L>T$ with $B(L)>\varepsilon L/(\log L)^{1/2}$ (display (16)); by Theorem 2 this gives Theorem 4. For (16), $cn<N(n)\le\sum_iB(n/b_i)$ (display (17)) is split by the size of $b_i$ against $T$ and $n/T$; if (16) failed for every $L>T$, each part would be at most a small multiple of $n$ (displays (21) and (23)), a contradiction for $\varepsilon$ small. The paper says this follows Raikov's method without using his theorem.

Dependencies

Theorem 2.

Bears on

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