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Statement

Theorem XIII (p. 18). Let $f$ be a real additive function with $f(m+1)-f(m)\to0$ as $m\to\infty$. Then $f(m)=c\log m$ for a constant $c$.

The paper adds (p. 19) that the conclusion seems likely to hold under $\frac1n\sum_{m=1}^n|f(m+1)-f(m)|\to0$.

Proof pointer

Pp. 18--19. With $P_1<P_2<\cdots$ the prime powers and $c=\limsup f(P_i)/\log P_i$, the proof treats in turn the cases where infinitely many primes, finitely many but some, and no primes have a power $Q$ with $f(Q)/\log Q>c$, and then $f(Q)/\log Q<c$ for some $Q$; in each it constructs pairs of integers a bounded distance apart whose $f$-values differ by more than a fixed $\delta>0$, contradicting $f(m+1)-f(m)\to0$. On p. 3 the paper says it deduces this result from Theorem V; the written argument on pp. 18--19 does not invoke it.

Read depth

Claims checked: the statement and its proof on pp. 18--19 read on the page images for structure. Nothing here is independently reviewed.

Dependencies

None in the corpus.

Source. P. Erdős, On the distribution function of additive functions, Ann. of Math. (2) 47 (1946), 1--20, doi:10.2307/1969031; the edition read is named on the source card.

Bears on

• Problem 491: the theorem gives $f(n)=c\log n$, the problem's conclusion with error term 0, for additive $f$ with $f(n+1)-f(n)\to0$, a hypothesis that implies the problem's bounded differences.
• Problem 1122: related only: the theorem's hypothesis $f(n+1)-f(n)\to0$ differs from the problem's and does not imply it, since $f(n)=-\log n$ satisfies it and decreases at every $n$.