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Statement

Setting (p. 530). A function $f$ defined for non-negative integers $m$ is additive when $f(m_1m_2)=f(m_1)+f(m_2)$ whenever $(m_1,m_2)=1$, and $\phi$ is multiplicative when $\phi(m_1m_2)=\phi(m_1)\phi(m_2)$ whenever $(m_1,m_2)=1$. The paper assumes throughout $f(m)\ge0$ and $\phi(m)\ge1$, and notes that $\log\phi$ is additive when $\phi$ is multiplicative, so it treats additive functions only. $G(f,n)$ is the number of integers $m\le n$ with $f(m+1)\ge f(m)$, and $S(f,n)$ the number with $f(m+1)\le f(m)$.

Theorem (p. 530, quoted). "Let the additive function $f(m)\geqslant0$ satisfy the following condition: $\sum\frac{f(p)}{p}$ converges when the summation is extended to all primes $p$. Then"

$$\lim_{n\to\infty}\frac{G(f,n)}{n}=\tfrac12,\qquad(1)$$ $$\lim_{n\to\infty}\frac{S(f,n)}{n}=\tfrac12.\qquad(2)$$

What the paper proves (p. 530) is that $G(f,n)/S(f,n)\to1$ and that the number of $m\le n$ with $f(m+1)=f(m)$ is $o(n)$; since $G(f,n)+S(f,n)$ is $n$ plus that number, (1) and (2) follow. In particular the integers $m$ with $f(m+1)>f(m)$, and those with $f(m+1)<f(m)$, each have density $\tfrac12$.

Extension (pp. 534--535). The paper states that the same theorem holds, and can be proved in a similar way, when $\sum_pf(p)/p$ diverges but the primes split into two classes $q_1$ and $q_2$ such that both $\sum_{q_1}f(q_1)/q_1$ and $\sum_{q_2}1/q_2$ converge. No proof is written out.

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, the statement on p. 530, the proof on pp. 530--534, the extension on pp. 534--535. The edition read is identified on the source card.

Read depth. Claims checked: the setting and the statement 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. 531--534. The paper first treats the case $f(p^\alpha)=f(p)$ for all $\alpha$ and truncates to $f_k(m)=\sum_{p\mid m,\,p\le p_k}f(p)$, with $p_k$ the $k$-th prime. Writing $a(m)$ for the largest squarefree divisor of $m$ built from primes up to $p_k$, the counts of $m\le n$ with $a(m)=a_i$ and $a(m+1)=a_j$ are estimated by the sieve of Eratosthenes, (3), and are asymptotically symmetric in $a_i,a_j$, (4), which gives $G(f_k,n)/S(f_k,n)\to1$. Lemma 1 (p. 532) bounds the number of near-ties $|f_k(m+1)-f_k(m)|\le\delta$ by $\tfrac12\varepsilon n$ for $k>k(\varepsilon)$, and Lemma 2 (p. 533) bounds by $\tfrac12\varepsilon n$ the number of $m\le n$ with $f(m)-f_k(m)>\delta$ or $f(m+1)-f_k(m+1)>\delta$, using the convergence of $\sum f(p)/p$. Together they give (5) and (6), $|G(f,n)-G(f_k,n)|<\varepsilon n$ and $|S(f,n)-S(f_k,n)|<\varepsilon n$, and, by the same split, the $o(n)$ bound for ties. The general case $f(p^\alpha)\ne f(p)$ is outlined on p. 534, using that at most $c_{10}n/p_k$ integers $m\le n$ are divisible by a square above $p_k$. The method is that of Erdős's paper "On the density of some sequences of numbers", J. London Math. Soc. 10 (1935), 120--125, whose Lemma 1 is used in the proof of Lemma 1 here.

Dependencies

None in the corpus. External input: Lemma 1 of Erdős, J. London Math. Soc. 10 (1935), 120--125, as cited on p. 533.

Bears on

No problem directly. The theorem bears on Problem 415 only through its application to Euler's function on the p. 534 consequence, whose page states the relation.