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Statement

Setting (p. 535). $V(m)$ is the number of distinct prime factors of $m$. $G(V,n)$ is the number of integers $m\le n$ with $V(m)\le V(m+1)$, and $S(V,n)$ the number with $V(m)\ge V(m+1)$.

(9) and (10) (p. 535, concluded p. 539). The paper does not call this a theorem; it states "We prove that" (9) and (10).

$$\lim_{n\to\infty}\frac{G(V,n)}{n}=\tfrac12,\qquad \lim_{n\to\infty}\frac{S(V,n)}{n}=\tfrac12.$$

The paper proves $G(V,n)/S(V,n)\to1$ and that only $o(n)$ integers $m\le n$ have $V(m)=V(m+1)$ (p. 539), from which (9) and (10) follow. So the integers $m$ with $V(m+1)>V(m)$, and those with $V(m+1)<V(m)$, each have density $\tfrac12$.

$V$ is additive with $V(p)=1$, so $\sum_pV(p)/p$ diverges and the Theorem of p. 530 does not apply. The paper notes (p. 535) that its method with a fixed truncation gives nothing, because Lemma 2 breaks down, and takes the truncation point $k$ as a function of $n$, for example $k=n^{1/(\log\log n)^2}$.

Source. P. Erdős, On a problem of Chowla and some related problems, Proc. Cambridge Philos. Soc. 32 (1936), 530--540, doi:10.1017/S0305004100019277: Section 2, the statement on p. 535, the proof on pp. 535--539. The edition read is identified on the source card.

Read depth. Claims checked: the setting, (9), (10) and the conclusion on p. 539 were read clause by clause on the printed pages. The proof was followed for structure and not verified. Nothing here is independently reviewed.

Proof pointer

Pp. 535--539. With $V_k(m)$ the number of distinct primes not greater than $p_k$ dividing $m$ (the paper later writes $k$ for this bound), the paper proves (11), $G(V_k,n)/S(V_k,n)\to1$, by estimating the number of $m\le n$ with $a(m)=a_i$, $a(m+1)=a_j$ through Brun's method and Landau's form of the sieve, giving the near-symmetry (15). Lemma 3 (p. 538) shows that $|V_k(m+1)-V_k(m)|<(\log\log\log n)^4$ holds for only $o(n)$ integers $m\le n$, and Lemma 4 (p. 539) that $V(m)-V_k(m)>(\log\log\log n)^2$ or $V(m+1)-V_k(m+1)>(\log\log\log n)^2$ holds for only $o(n)$ of them. As in Section 1 these give (16) and (17), $|G(V,n)-G(V_k,n)|<\varepsilon n$ and $|S(V,n)-S(V_k,n)|<\varepsilon n$, and the $o(n)$ bound for ties.

Dependencies

The method of the Theorem of p. 530.

Bears on

No problem directly. (9) and (10) are the step from which the paper derives Chowla's conjecture.