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Claim. Erdős proves that a real additive function $f$ equals $c\log n$ for a constant $c$ in two cases: when $f(n+1)\ge f(n)$ for every $n$ (Theorem XI), and when $f(n+1)-f(n)\to0$ (Theorem XIII). For Theorem XI the monotonicity gives $f(m)\le f(n)\le f(2)+f(m)$ for odd $m<n<2m$, so $f$ is finitely distributed in the paper's sense and Theorem V writes $f(m)=c\log m+\varphi(m)$ with $\sum_p\varphi'(p)^2/p$ convergent; Theorem X, on the distribution of $f(m+1)-f(m)$, then forces $\varphi$ to vanish. The paper omits the proof of Theorem X as similar to that of an earlier paper. Theorem XIII is proved directly, by comparing $f$ at integers built from the prime powers on which $f(Q)/\log Q$ approaches its upper limit. The same paper states the question of Problem 491 itself, that bounded consecutive differences give $f(n)=c\log n+\varphi(n)$ with $\varphi$ bounded, as a result that probably holds but that Erdős cannot prove.

Covers. The instances of Problem 491 in which $f$ is nondecreasing or $f(n+1)-f(n)\to0$. Both classes have bounded consecutive differences, and for them the answer is yes with $f(n)=c\log n$ exactly. General bounded differences are settled by Wirsing's theorem.

Depends on. No page of this wiki.

Acceptance. Refereed: P. Erdős, On the distribution function of additive functions, Ann. of Math. (2) 47 (1946), no. 1, 1–20, received 24 February 1945, which is the date the page carries; the paper is digested on the library's card. The site's commentary credits the two cases to this paper, but its PROVED label credits Wirsing, so no reviewed evidence is listed. No Lean checks these statements, so no formalized evidence is listed.