Problem 1122

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claims/: The 3 claim pages of Problem 1122, one per claimant's result; the problem's standing derives from them.

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Statement. Let $f:\mathbb{N}\to \mathbb{R}$ be an additive function (i.e. $f(ab)=f(a)+f(b)$ whenever $(a,b)=1$). Let

$$A=\{ n \geq 1: f(n+1)< f(n)\}.$$

If $\lvert A\cap [1,X]\rvert =o(X)$ then must $f(n)=c\log n$ for some $c\in \mathbb{R}$?

Status. Claimed, proved: no claim settling the problem is accepted. The site labels the problem OPEN (page last edited 2026-04-01). One pending full claim is on its proof-claims tab, recorded and not adopted: Gu 2026, a manuscript posted on Zenodo on 2026-09-08 (the record states publication date 2026-09-07) and registered on the site the same day, which asserts the answer yes with a nonnegative constant $c$ and credits GPT-6 Astra for the argument. It comes with a Lean development that derives the statement from four cited theorems taken as hypotheses; no review or acceptance of the claim is recorded. Two refereed partial results that the site's commentary credits are accepted: Erdős 1946 [Er46], the answer yes when $A$ is empty or when $f(n+1)-f(n)\to0$, and Mangerel 2021 [Ma22], the answer yes for completely additive $f$ with no outsized prime values whose decreases number $\ll X/(\log X)^{2+\delta}$.

Source. erdosproblems.com/1122, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #1122, https://www.erdosproblems.com/1122.

References.

• [Er46] Erdős, P., On the distribution function of additive functions. Ann. of Math. (2) 47 (1946), 1-20.
• [Ma22] Mangerel, Alexander P., Additive functions in short intervals, gaps and a conjecture of Erdős. Ramanujan J. (2022), 1023-1090.

Formalization. None recorded: no formal-conjectures statement file exists for the problem and the community database lists no formal statement; the Lean development posted with the Gu claim is linked from its claim page.

Current assessment

The question (site formulation, page last edited 2026-04-01). Whether an additive $f:\mathbb{N}\to\mathbb{R}$ whose decreases $A=\{n: f(n+1)<f(n)\}$ satisfy $|A\cap[1,X]|=o(X)$ must equal $c\log n$ for some real $c$. The site labels the problem OPEN.

Standing. Claimed, proved: the full claim Gu 2026, a manuscript posted on Zenodo on 2026-09-08 (the record states publication date 2026-09-07) and registered on the site's proof-claims tab the same day, asserts the answer yes with $c\ge0$ and credits GPT-6 Astra for the argument; no review or acceptance of it is recorded. Two partial claims are accepted on refereed evidence. Erdős [Er46], in the paper that poses the conjecture, proved that $f(n)=c\log n$ when $f(n+1)\ge f(n)$ for every $n$, that is, when $A$ is empty, and also when $f(n+1)-f(n)\to0$; Erdős 1946 covers the instances with $A$ empty, which have density zero trivially. The other is Mangerel 2021 (Corollary 1.7 of [Ma22], Ramanujan J. 59 (2022)): for completely additive $f$ with $\lim_{\varepsilon\to0^+}F_f(\varepsilon)=0$ and $|A\cap[1,X]|\ll X/(\log X)^{2+\delta}$ for some $\delta>0$, the answer is yes. The site's commentary states that hypothesis as additive, which overstates the corollary; it credits the partial progress but labels the problem OPEN, so no reviewed evidence is listed.

Lean coverage. No formal-conjectures statement file exists for the problem and the community database lists no formal statement. The Lean development posted with the Gu claim derives the statement from four cited theorems taken as hypotheses; the corpus has not built or audited it, so no formalized evidence is listed.

Search scope. The site's problem page (last edited 2026-04-01) and its proof-claims thread, as of 2026-10-06; the Zenodo record and archive of the Gu claim, as of 2026-10-07; and the library cards of [Er46] and [Ma22]. The proofs are not compiled in this wiki.

Linked library material

These entries are derived from explicit links on library pages. They are navigation only and do not by themselves record mathematical progress.

• erdos_1946_distribution_function_additive_functions
• erdos_1946_distribution_function_additive_functions / conjecture_p3
• erdos_1946_distribution_function_additive_functions / theorem_11
• erdos_1946_distribution_function_additive_functions / theorem_13
• mangerel_2022_additive_functions_short_intervals_gaps_conjecture