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Statement

Consequence (p. 534). Here $\sigma(m)$ is the sum of the divisors of $m$ and $\phi(m)$ is Euler's function. Applying the Theorem of p. 530 to the functions $\sigma(m)/m$ and $m/\phi(m)$, the paper deduces that the number of integers $m\le n$ with $\sigma(m+1)>\sigma(m)$ is asymptotically $\tfrac12n$, and states that "the same is true for $\phi(m)$" (p. 534), that is, the number of $m\le n$ with $\phi(m+1)>\phi(m)$ is asymptotically $\tfrac12n$.

The deduction rests on the paper's further statement (p. 534) that Lemmas 1 and 2 give only $o(n)$ integers $m\le n$ for which the sign of $\sigma(m)/m-\sigma(m+1)/(m+1)$ differs from the sign of $\sigma(m)-\sigma(m+1)$. The paper writes this out for $\sigma$ only and asserts the $\phi$ case without further detail. It states the count for $\phi(m+1)>\phi(m)$; it does not separately state the count for the reverse inequality.

The paper takes $f$ to be these multiplicative functions, which the Theorem covers through the reduction of p. 530: for multiplicative $\phi\ge1$, $\log\phi$ is additive. The paper does not check the hypothesis; it holds because the values of $\sigma(m)/m$ and $m/\phi(m)$ at a prime $p$ are $1+1/p$ and $p/(p-1)$, whose logarithms are of order $1/p$.

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 1, p. 534. The edition read is identified on the source card.

Read depth. Claims checked: the statement was read clause by clause on the printed page. The sign comparison is asserted in the paper with a one-line justification and was not verified. Nothing here is independently reviewed.

Proof pointer

P. 534. The Theorem gives density $\tfrac12$ for $\sigma(m+1)/(m+1)>\sigma(m)/m$; the sign comparison transfers this to $\sigma(m+1)>\sigma(m)$, with $o(n)$ exceptions.

Dependencies

The Theorem of p. 530 and its Lemmas 1 and 2 (pp. 532--533).

Bears on

• Problem 415: the problem asks about the ordering patterns of $k$ consecutive values of $\phi$. For $k=2$ the paper states that $\phi(m+1)>\phi(m)$ holds for asymptotically half of the integers $m\le n$. It says nothing about the growth of $F(n)$ or about longer patterns.