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Claim. Let $g:\mathbb{N}\to\mathbb{R}$ be completely additive, let $B_g(X)=(\sum_{p^k\le X}g(p^k)^2/p^k)^{1/2}$, and let

$$F_g(\varepsilon)=\limsup_{X\to\infty}\frac{1}{B_g(X)^2} \sum_{\substack{p\le X\\ |g(p)|>\varepsilon^{-1}B_g(X)}}\frac{g(p)^2}{p},$$

the share of $B_g(X)^2$ carried by primes where $|g(p)|$ exceeds $\varepsilon^{-1}B_g(X)$. If $F_g(\varepsilon)\to0$ as $\varepsilon\to0^+$ and the set $\mathcal{B}=\{n: g(n)<g(n-1)\}$ of decreases satisfies $|\mathcal{B}\cap[1,X]|\ll X/(\log X)^{2+\delta}$ for some $\delta>0$, then $g(n)=c\log n$ for all $n$ and some constant $c$. This is Corollary 1.7 of Mangerel, Additive functions in short intervals, gaps and a conjecture of Erdős, Ramanujan J. 59 (2022), no. 4, 1023–1090, digested on the card Mangerel 2022. The paper derives it from its Theorem 1.8, that an additive function in the class $\mathcal{A}_s$ (additive functions with $B_g(X)\to\infty$, whose prime values dominate $B_g(X)^2$, and with $F_g(\varepsilon)\to0$) whose decreases have density zero is close to $\lambda(X)\log$ at prime powers, together with a growth estimate $B_g(X)=(\log X)^{1+o(1)}$ proved in its Section 7.2; the main tools are the paper's short-interval averaging theorems for additive functions, analogs of the Matomäki–Radziwiłł theorem.

Covers. The instances of Problem 1122 in which $f$ is completely additive with $\lim_{\varepsilon\to0^+}F_f(\varepsilon)=0$ and the set $A$ of decreases satisfies $|A\cap[1,X]|\ll X/(\log X)^{2+\delta}$ for some $\delta>0$: for these the answer is yes. It leaves open additive functions that are not completely additive, functions with $F_f(\varepsilon)\not\to0$, and the full hypothesis $|A\cap[1,X]|=o(X)$; the site's commentary states the hypothesis as additive, which overstates the corollary. Problem 1122 as a whole has the pending full claim Gu 2026, which cites this corollary as prior work, and the earlier accepted partial claim Erdős 1946 settles the instances with no decreases at all.

Depends on. No page of this wiki: the proof is self-contained in the paper.

Acceptance. Refereed: the paper appeared in The Ramanujan Journal, volume 59, issue 4 (2022), pages 1023–1090, published online 2022-09-03. The site's commentary (page last edited 2026-04-01) credits the partial progress to this paper as [Ma22], but the site labels the problem OPEN, so no reviewed evidence is listed. No Lean checks the statement, so no formalized evidence is listed. The page is dated by the first arXiv version, submitted 2021-08-27. The proof is not compiled in this wiki.