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Statement

Chowla's conjecture (stated p. 530, proved p. 540). With $d(m)$ the number of divisors of $m$, the integers $m$ for which $d(m+1)>d(m)$ have density $\tfrac12$. The paper's opening (p. 530) records the conjecture as Chowla's and announces its proof in Section 2.

The paper reduces the conjecture (p. 540) to showing that only $o(n)$ integers $m\le n$ have $V(m+1)-V(m)\ge0$ and $d(m+1)-d(m)\le0$, or $V(m+1)-V(m)\le0$ and $d(m+1)-d(m)\ge0$, where $V(m)$ is the number of distinct prime factors of $m$; combined with (9) and (10) this gives the density $\tfrac12$. The paper does not state the reverse case, though the same reduction gives density $\tfrac12$ for $d(m+1)<d(m)$.

Footnote theorem (p. 540). The paper states: for any function $X(n)$ with $X(n)\to\infty$, for almost all integers $m\le n$, as printed,

$$\frac{\log\log n}{X(n)}<|V(m+1)-V(m)|<\log\log n\,X(n).$$

It says the first inequality may be proved by lemmas similar to but stronger than Lemmas 3 and 4, and gives no proof of it. For the second it reproduces P. Turán's argument, which bounds $\sum_{m\le n}(V(m+1)-V(m))^2$ by $O(n\log\log n)$.

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

Read depth. Claims checked: the statement, the reduction and the footnote were read clause by clause on the printed pages. The proof on p. 540 was followed and not independently verified. Nothing here is independently reviewed.

Proof pointer

P. 540. By Lemmas 3 and 4, $|V(m+1)-V(m)|>(\log\log\log n)^2$ for almost all $m\le n$. Among the $m$ with $V(m+1)-V(m)\ge0$ and $d(m+1)\le d(m)$, those with $V(m+1)-V(m)<(\log\log\log n)^2$ are therefore $o(n)$. The others satisfy $d(m)\ge d(m+1)\ge2^{V(m+1)}\ge2^{V(m)}2^{(\log\log\log n)^2}$; writing $m=AB^2$ with $A$ squarefree gives $d(m)\le2^{V(m)}d(B^2)$, so $B^2$ is at least $2^{(\log\log\log n)^2}$, and the number of $m\le n$ divisible by such a square is $o(n)$.

Dependencies

(9), (10) and Lemmas 3 and 4 of the same paper.

Bears on

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