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Statement

Lemma 1 (p. 109). Let $q_1,q_2,\ldots$ be primes with $\sum_{i=1}^\infty 1/q_i=\infty$, and let $v_q(n)$ be the number of the $q_i$ dividing $n$. Then for every $A$ the integers $n$ with $v_q(n)<A$ have density 0.

Source. P. Erdős, On amicable numbers, Publ. Math. Debrecen 4 (1955), 108--111: Lemma 1 on p. 109. The edition read is identified on the source card.

Read depth. Claims checked: the statement and the paper's derivation from Turán's theorem were read clause by clause on p. 109. Turán's theorem itself was not checked here. Nothing here is independently reviewed.

Proof pointer

P. 109. The paper gives no separate proof: the lemma is a special case of a theorem of Turán (J. London Math. Soc. 11 (1936), 125--133), which the paper quotes in a weaker form as follows. If $0\le\psi(p)\le K$ for all primes $p$, $\sum_p\psi(p)/p=\infty$, and $\psi(n)=\sum_{p\mid n}\psi(p)$ over the distinct prime factors of $n$, then for all but $o(N)$ of the $n\le N$

$$\Bigl|\psi(n)-\sum_{p\le N}\frac{\psi(p)}{p}\Bigr| <\Bigl(\sum_{p\le N}\frac{\psi(p)}{p}\Bigr)^{3/4}.\qquad(1)$$

Lemma 1 takes $\psi(p)=1$ for $p$ in the sequence and $\psi(p)=0$ otherwise, so that $\psi(n)=v_q(n)$ and the main term in (1) tends to infinity.

Dependencies

Turán's theorem cited above, an external input not recorded in the corpus.

Bears on

No problem directly. The lemma enters the proof of the theorem through Lemma 2, and bears on Problem 830 only through it; the theorem's page states the relation.