Erdős #50 — no point of positive derivative for the φ(n)/n distribution

Statement

Schoenberg proved that for every $c\in[0,1]$ the density \[f(c) = \operatorname{dens}\{n\in\mathbb{N} : \phi(n) < cn\}\] exists (φ = Euler's totient), giving a continuous, strictly increasing distribution function $f$ on $[0,1]$. Erdős proved $f$ is purely singular (continuous, $f'=0$ a.e., not absolutely continuous). Open question: is it true that there is no $x$ for which $f'(x)$ exists and is positive? (erdosproblems.com/50)

Facts

- Prize $250; status open, explicitly "cannot be resolved with a finite computation" (erdosproblems.com/50) — the claim is a statement about every real $x\in[0,1]$, so no finite check settles it either way. - Falsifiable: no in the finite-computation sense. A single explicit exceptional point $x_0$ with $f'(x_0)$ existing and positive would refute it, but "exhibiting" such a point analytically (if one exists) is itself a hard analytic construction, not a search; conversely proving the claim needs a genuine "for all $x$" argument. - Origin: Erdős [Er95, "Some of my favourite problems in number theory, combinatorics and geometry", Resenhas 4 (1995), 165–186] states the open question; the underlying singularity result is Erdős's own 1939 paper, "On the smoothness of the asymptotic distribution of additive arithmetical functions", Amer. J. Math. 61 (1939), 722–725 (confirmed via Erdős's own publication list at ftp.gwdg.de/pub/misc/EMIS/classics/Erdos/multnum.htm, and independently cited as such by arXiv:2603.11196). - Known results / best bounds: - The existence of the limit distribution $f$ itself is due to I. J. Schoenberg (classical, pre-Erdős; erdosproblems.com/50 attributes it directly to him). - Erdős (1939) proved $f$ is purely singular continuous — this already forces $f'=0$ Lebesgue-almost-everywhere, but says nothing about the (measure-zero) exceptional set the open problem asks about. - The sibling distribution of $\sigma(n)/n$ (sum-of-divisors) is also known to be purely singular, per Erdős, cited in the 2010-era paper "The distribution functions of $\sigma(n)/n$ and $n/\phi(n)$, II" (arXiv:1011.4262, read via fetch) — same phenomenon, same open-style fine-structure gap, no full resolution found there either. - Most recent relevant progress (2026): arXiv:2603.11196 (V. S. Sehrawat) proves, for a structurally analogous random-Euler-product measure $\mu_G$ (arising from primitive-root densities over $\mathbb F_p$, built the same way as $\phi(n)/n$'s limit law — as an infinite product $\prod_\ell (1-1/\ell)^{B_\ell}$ over independent Bernoulli digits $B_\ell$), that $\mu_G$ is singular with Hausdorff dimension zero ($\dim_H\mu_G=0$) and that $\log\mu_G$ is a Rajchman measure (Fourier coefficients $\to0$) with an explicit polynomial decay rate under a Graham–Kolesnik exponential-sum envelope. The paper's own abstract states this result "sharpen[s] Erdős (1939) for $\phi(n)/n$" — i.e. the state of the art for this *exact family* of problems has moved from "singular" (1939) to "singular with dimension-zero support + quantitative Fourier decay" (2026), via Fourier-analytic / exponential-sum methods rather than the original combinatorial method. This is the closest genuinely new machinery found; it does not itself resolve the "no positive-derivative point" question for $\phi(n)/n$ (it addresses the shifted-prime/primitive-root analogue, not $\phi(n)/n$ itself), but it is the most directly transplantable modern technique. - No paper found (arXiv, OpenAlex-style search, Google Scholar via WebSearch) that proves or disproves the existence of an exceptional point with $f'(x)>0$ finite for the $\phi(n)/n$ distribution specifically, or that even conjectures the answer. - Related problems: no "See also" cross-links are given on erdosproblems.com/50 itself, and the topically-adjacent problems #48 (φ(n)=σ(m) solvability, resolved by Ford–Luca–Pomerance/Garaev) and #49 (longest φ-increasing subsequence, resolved by Tao 2024) are about *value-set* questions, not distribution-singularity questions, so they are not real technical analogs despite sharing the "φ, σ" tag cluster.

Literature state

Not resolved anywhere. The literature search (arXiv, Google/Scholar-style web search, the erdosproblems.com bib/forum, and a related-paper on σ(n)/n) confirms: 1. The base fact Erdős himself established — pure singularity of $f$ — dates to 1939 and is well-cited (Amer. J. Math. 61, 722–725); no stronger statement about $f$'s *pointwise* derivative behavior beyond a.e.-zero is claimed anywhere I could find for $\phi(n)/n$ itself. 2. The nearest *technical* progress is not on $\phi(n)/n$ but on a structurally identical construction: arXiv:2603.11196 (2026, V. S. Sehrawat) pushes an analogous singular measure (built from an independent-Bernoulli infinite Euler product, exactly the probabilistic skeleton underlying $\phi(n)/n$'s limit law via the Erdős–Kac/Landau heuristic "$p\mid n$ with heuristic independence and density $1/p$") to Hausdorff dimension zero and Rajchman/Fourier-decay status, using Fourier-analytic exponential-sum estimates rather than Erdős's original method. This shows the modern toolkit (characteristic-function / Fourier-decay analysis of infinite independent-digit products) is the currently active line of attack on this whole problem family, and that it can produce results *strictly stronger* than plain singularity (dimension-zero support, quantitative decay) — but dimension-zero support alone does not settle whether points *off* that thin support could still have $f'(x)$ finite and positive; ruling that out needs a separate "0/∞ (or 0/undefined) dichotomy at every point" argument, not just an a.e. or dimension statement. 3. arXiv:1011.4262 confirms the same open-ended situation holds for the sibling function $\sigma(n)/n$ (also proved singular by Erdős), suggesting the "no positive-derivative point anywhere" question is a genuine unsolved sub-problem of the broader theory of distribution functions of multiplicative functions (Erdős–Wintner / Elliott's probabilistic number theory framework), not a quirk specific to $\phi$. 4. hal.science/hal-00871083 ("Distribution functions of the sequence $\phi(n)/n$, $n\in(k,k+N]$") appears (via WebSearch snippet only, direct fetch blocked by bot-protection) to extend Schoenberg/Erdős's results to short intervals but its content on the derivative question could not be verified directly — not relied on for any claim above. 5. No AI-assisted attempt log (GitHub "AI contributions" style tracker, Lean formalization repos, LLM-hunter trackers) was found specifically addressing problem #50, unlike some other Erdős problems (contrast with e.g. #40, which has a DeepMind Lean formalization and a public LLM attempt log — neither exists here as far as this search found).

Attack surface

- Mode: derivation+formalization (explicitly "not finite" per the site; requires a genuine "for all $x\in[0,1]$" analytic argument, most plausibly via the 0/∞-style dichotomy technique below). - Concrete first experiment: (1) Make explicit the random-series representation underlying $f$: $-\log(\phi(n)/n) = -\sum_{p\mid n}\log(1-1/p)$ behaves, for $n$ ranging uniformly to $x$, like a sum of independent random variables $X_p \in \{0, -\log(1-1/p)\}$ with $\Pr(X_p\ne0)=1/p$ (the Erdős–Kac / Landau heuristic, rigorized via Erdős–Wintner-style sieve estimates) — i.e. $f$ is (up to a smooth reparametrization) the CDF of a Jessen–Wintner random series of exactly the type that Salem's singular functions and the Minkowski question-mark function are built from. (2) Adapt the technique of R. Salem (Trans. AMS 53 (1943), 427–439) and, in the continued-fraction setting, N. Moshchevitin–A. Dushistova / I. Gayfulin (arXiv:2107.00461) — who prove the *derivative of $?(x)$, when it exists, is either $0$ or $+\infty$, never a finite positive value* — to the $\phi(n)/n$ random-series representation: bound, at every point $x$ (not just a.e.), the local ratio $\log f(\text{interval around }x)/\log(\text{interval length})$ using a strong-law / large-deviation argument on the partial sums $\sum_{p\le P} X_p$, aiming to show this ratio can never converge to exactly $1$ (which is what a finite positive derivative would require) — it should generically diverge (dimension $<1$, giving $f'=0$) or, at genuinely exceptional points, blow up, but literally never sit at the borderline value $1$ with a two-sided finite nonzero limit. (3) As a numerical sanity/derivation-fuel step (not a proof): compute $f$ on a fine grid via $\phi(n)/n$ for $n$ up to $10^7$–$10^8$ and estimate local Hölder/scaling exponents $\log(f(x+h)-f(x-h))/\log(2h)$ at a battery of "suspicious" points (dyadic rationals, points where partial Euler-product truncations are exactly balanced) to see whether the exponent ever numerically approaches $1$ — this cannot prove the infinite statement but would flag whether the conjectured answer ("no such $x$") looks numerically robust or fragile. - Oracle: none for the actual open statement (it is a universally-quantified analytic claim, "not finite" per the site). The numerical Hölder-exponent scan in step (3) above is mechanically checkable/reproducible but only heuristic, not a proof or disproof. - Feasibility: hard but not obviously out of reach for a derivation attempt — unlike a from-scratch open problem, this one has a strikingly close, actively-worked 2026 analog (arXiv:2603.11196) using exactly the right machinery (independent-digit infinite-product measures, Fourier/exponential-sum decay) on a sibling construction, plus a fully worked-out solved template in a different but structurally identical setting (Minkowski $?(x)$'s $0/\infty$ derivative dichotomy, Salem 1943 + Moshchevitin–Dushistova + Gayfulin 2021). The realistic near-term contribution is not a full proof but (1) writing out the precise random-series representation of $f$ rigorously (translating the Erdős–Kac heuristic into an Erdős–Wintner-style rigorous statement for this specific function) and (2) attempting to transplant the Salem/Minkowski "0-or-∞, never finite-positive" dichotomy argument, checking where the transplant breaks (the two settings differ: $?(x)$'s random series comes from a self-similar continued-fraction dynamical system with a clean transfer operator, while $\phi(n)/n$'s comes from an independent-per-prime sieve with vanishing-density digits $1/p\to0$, which is the same shape as Sehrawat's primitive-root construction — so the more directly relevant template may in fact be arXiv:2603.11196's Fourier-decay approach, extended from "dimension zero" to a full pointwise dichotomy).

Related

- erdos/48 — φ(n)=σ(m) solvability (resolved, Ford–Luca–Pomerance/Garaev); same φ/σ tag cluster on erdosproblems.com but a value-set question, not a distribution-singularity question — noted for completeness, not a technical analog. - erdos/49 — longest φ-increasing subsequence (resolved by Tao 2024); same cluster, same caveat as above. - concept/singular-distribution-function — the central object: a continuous, strictly increasing $f$ with $f'=0$ a.e. but not absolutely continuous. - concept/jessen-wintner-purity — the law of pure type for sums of independent random series (Jessen–Wintner theorem, en.wikipedia.org/wiki/Jessen–Wintner_theorem): explains *why* $f$ must be purely one type (discrete/AC/singular) and is the natural rigorous home for the random-series representation of $\phi(n)/n$'s limit law. - concept/erdos-wintner-theorem — the probabilistic-number-theory criterion (three-series theorem analog) for when additive arithmetic functions have a limiting distribution; the framework Erdős's 1939 singularity proof sits inside. - concept/minkowski-question-mark-function — solved analog: the derivative of $?(x)$, when it exists, is provably only $0$ or $+\infty$ (Salem 1943; sharp continued-fraction threshold by Moshchevitin–Dushistova, arXiv:2107.00461 Gayfulin) — the exact "no finite positive derivative anywhere" template this problem is asking to replicate for $\phi(n)/n$. - concept/salem-singular-functions — R. Salem, Trans. AMS 53 (1943), 427–439: the original 0/∞-dichotomy construction technique for strictly-increasing singular functions built from independent-digit random series. - concept/bernoulli-convolutions — the broader family of infinite-convolution / infinite-independent-digit singular measures (Peres–Solomyak and successors); shares the Jessen–Wintner purity framework and the local-dimension/multifractal toolkit. - concept/local-dimension-multifractal-analysis — the technique (pointwise scaling exponent $\log\mu(B(x,r))/\log r$) needed to go from "singular a.e." to "no point has a specific forbidden exponent," i.e. from Erdős's 1939 result to the open #50 question. - concept/rajchman-measure-fourier-decay — the modern tool used in arXiv:2603.11196 (Sehrawat 2026) to sharpen an Erdős-1939-type singularity result to Hausdorff-dimension-zero for a structurally identical random-Euler-product measure; the most promising transplantable machinery found for #50.

PRIOR-ART SCAN (2026-07-03)

Verdict: HARD-OPEN (confirmed, no change from 2026-07-02 pass; this pass closes the one gap the prior scan flagged — the GitHub angle — and finds nothing that alters the status).

What's new vs. the 2026-07-02 note

- The erdosproblems.com/50 page itself carries a "Formalised statement? Yes" link the prior pass missed: github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/50.lean (fetched directly). This is exactly the kind of GitHub-repo lead the #64 false-novelty mistake teaches us to run down before claiming anything. Checked thoroughly — it is not a solution. The file states three theorems (erdos_50_schoenberg, erdos_50_singular, erdos_50) with @[category research solved] / @[category research open] tags but every proof body is sorry (Lean's unproven-placeholder keyword) — it is a bare statement formalization, no proof content, for all three including the two "solved" (in the literature, not in this repo) sub-facts. - Checked the repo's commit history for this file via the GitHub API (api.github.com/repos/google-deepmind/formal-conjectures/commits?path=...): only 2 commits ever — the original statement PR #2139 (Ralf Stephan, "using Claude + Opus for supervised formalization tasks", 2026-03-19) and a later assumption-fix PR #4028 (2026-05-27). Neither adds a proof; both are pure formalization edits. - Checked the tracking issue github.com/google-deepmind/formal-conjectures/issues/270 ("Erdős Problem 50: Differentiability of the phi(n)/n distribution") — closed, but closed *by the formalization PR*, not by any mathematical resolution. - Checked erdosproblems.com/history/50 (full revision history): the problem statement and remark text are byte-identical between the 2025-10-20 revision and the current (2026-07-03) version — zero content changes, confirming no new remark/comment has landed since well before the prior scan. - Checked the teorth/erdosproblems GitHub wiki page "AI contributions to Erdős problems" (fetched raw, all ~680 lines grepped for "50", "phi(n)/n", "totient"): zero rows for problem #50 anywhere in the AI-attempt-log tables (several other problems — 150, 350, 550, 650, 750, 1150 — have logged AI attempts/solutions; #50 has none). This directly confirms, via primary source rather than absence-of-search-hits, that no AI system (GPT, Gemini, Aristotle, Claude, etc.) has a logged attempt on this problem. - Re-confirmed arXiv:2603.11196 (Sehrawat) abstract directly via the arXiv API (not just WebFetch summary this time): title is "Primitive-root determinant densities: extremal order, dimension-zero, Fourier decay, and a lattice-smoothing no-go"; abstract explicitly states $\mu_G$ is singular with $\dim_H\mu_G=0$, "sharpening Erdős (1939) for $\varphi(n)/n$" — consistent with the prior pass, still the closest transplantable technique, still does not touch the pointwise-derivative question. - New context found (not previously cited): three general (non-φ(n)/n-specific) papers on singular functions and pointwise derivatives — arXiv:2003.06338 (Kahane-style construction: an increasing singular $f$ can be built with $f'(x)=g(x)$ prescribed on any countable set $A$), arXiv:2111.14519 (Sánchez–Viader–Paradís–Carrillo-style result: the set of points where a singular function has nonzero finite derivative can be made to contain any prescribed $F_\sigma$ null set), and arXiv:2201.01521 (law-of-pure-types for CDFs of stationary-digit expansions, a relative of the Jessen–Wintner framework). Net effect on the problem: these confirm that "singular" alone never generically implies "no point of positive finite derivative" — some singular functions constructed by hand *do* have such points — so Erdős's #50 question is a genuine structure-specific fact about $\varphi(n)/n$'s particular random-Euler-product construction, not something a generic singular-function theorem could settle either way. This reinforces (does not overturn) the existing HARD-OPEN verdict and the Salem/Minkowski-?(x) transplant as the most concrete attack. - Re-ran targeted WebSearch queries ("Erdos problem 50 phi(n)/n distribution derivative positive 2026", "Erdos problem 50 erdosproblems.com/50 solved OR partial result OR arXiv", "Schoenberg distribution function totient singular derivative pointwise exceptional set arXiv 2024 2025 2026"): no new paper, preprint, or forum post proving, disproving, or claiming partial progress on the $\varphi(n)/n$-specific question was found in any of them.

Sources actually read this pass

erdosproblems.com/50 (raw HTML, direct curl), erdosproblems.com/history/50, github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/50.lean (raw file), api.github.com/repos/google-deepmind/formal-conjectures/commits?path=FormalConjectures/ErdosProblems/50.lean, github.com/google-deepmind/formal-conjectures/issues/270, api.github.com/search/issues?q=repo:google-deepmind/formal-conjectures+erdos+50+phi, raw.githubusercontent.com/wiki/teorth/erdosproblems/AI-contributions-to-Erdős-problems.md, arXiv API abstracts for 2603.11196, 2111.14519, 2201.01521, 2003.06338.

No concrete first experiment added — this is a confirmation pass, not a fresh attack plan; the "Attack surface" section above (Salem/Minkowski-?(x) 0-or-∞ dichotomy transplant + Sehrawat Fourier-decay machinery + numerical Hölder-exponent scan) from the 2026-07-02 pass stands unchanged as the best available concrete plan.