---

Claim. Let $f:\mathbb{N}\to\mathbb{R}$ be additive. If $f(n+1)\ge f(n)$ for every $n$, or if $f(n+1)-f(n)\to0$, then $f(n)=c\log n$ for some constant $c\in\mathbb{R}$. Both statements are proved in P. Erdős, On the distribution function of additive functions, Ann. of Math. (2) 47 (1946), 1--20, digested on the card Erdős 1946; the paper also poses the question of Problem 1122 as a conjecture, and Mangerel restates the two results in Section 1.2 of the paper digested on Mangerel 2022. The monotone case is the one the site's commentary describes as the case of an empty $A$: the set $A=\{n: f(n+1)<f(n)\}$ of decreases is empty exactly when $f$ is nondecreasing. The argument rests on the paper's structural theorem that an additive function whose values on a positive proportion of the integers up to $n$ lie within a bounded distance of each other is close to $c\log$ at the primes, in the sense that $\sum_p((f(p)-c\log p)/p)^2$ converges for some $c$.

Covers. The instances of Problem 1122 in which $A$ is empty, that is, $f$ is nondecreasing: an empty set has density zero, and for these the answer is yes. Also the functions with $f(n+1)-f(n)\to0$, for which the conclusion holds whether or not $A$ has density zero. It leaves open every $f$ with a nonempty density-zero set of decreases whose differences do not tend to zero, which is the problem's content; the refereed partial claim Mangerel 2021 covers some of these, and the pending full claim Gu 2026 asserts the rest.

Depends on. No page of this wiki: the proof is the paper's own.

Acceptance. Refereed: the paper appeared in the Annals of Mathematics, second series, volume 47, number 1 (January 1946), pages 1--20; the page is dated by the issue's nominal first day. The site's commentary (page last edited 2026-04-01) credits the two results to Erdős as [Er46], but the site labels the problem OPEN, so no reviewed evidence is listed. No Lean checks the statement, so no formalized evidence is listed. No file of the paper is held; the card cites the edition it names. The proof is not compiled in this wiki.