[#P2998] Non-existence of positive derivatives for the distribution function of Euler's totient ratio

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Problem. Erdős Problem 50: Let $\varphi$ denote Euler's totient function. A function $f : \mathbb{R} \to \mathbb{R}$ is called the asymptotic distribution function of $\varphi(n)/n$ if for every $c \in [0, 1]$, the set $\{n \in \mathbb{N} : \varphi(n) < cn\}$ has a natural density, and this density equals $f(c)$. Determine whether the following holds: for every function $f : \mathbb{R} \to \mathbb{R}$ that is the asymptotic distribution function of $\varphi(n)/n$, there do not exist $x \in [0, 1]$ and $y > 0$ such that $f$ has derivative $y$ at $x$ within the interval $[0, 1]$.

A finite mathematical diagram showing sample totient ratios arranged as a distribution function. — Sample totient ratios arranged as a distribution curve.

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Definitions and notation

1Context

This problem concerns the fine analytic properties of the distribution function of values of $\varphi(n)/n$, where $\varphi$ is Euler's totient function. Schoenberg proved that this distribution function exists, and Erdős proved that it is purely singular continuous (continuous with derivative zero almost everywhere). The question asks whether the derivative can be positive at any point, or equivalently whether the function is nowhere differentiable with positive derivative.

2Problem setup

Remark 1. Euler's totient function $\varphi(n)$ counts the number of integers in $\{1, 2, \ldots, n\}$ that are relatively prime to $n$.

Definition 1 (The natural density of a set $A \subseteq \mathbb{N}$). The natural density of a set $A \subseteq \mathbb{N}$ is $\lim_{N \to \infty} \frac{|A \cap \{1, 2, \ldots, N\}|}{N}$, when this limit exists.

Remark 2. A function $f$ has derivative $y$ at $x$ within a set $S$, denoted by a suitable derivative notion, if the difference quotient of $f$ at $x$ converges to $y$ when approaching $x$ through points of $S$.

3What counts as a solution

Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 50: Let $\varphi$ denote Euler's totient function. A function $f : \mathbb{R} \to \mathbb{R}$ is called the asymptotic distribution function of $\varphi(n)/n$ if for every $c \in [0, 1]$, the set $\{n \in \mathbb{N} : \varphi(n) < cn\}$ has a natural density, and this density equals $f(c)$. Determine whether the following holds: for every function $f : \mathbb{R} \to \mathbb{R}$ that is the asymptotic distribution function of $\varphi(n)/n$, there do not exist $x \in [0, 1]$ and $y > 0$ such that $f$ has derivative $y$ at $x$ within the interval $[0, 1]$.

1Status

Saved packet · July 31, 2026

What counts as a solution

Saved packet status (Current status and unresolved remainder). OPEN as of 2026-07-31. The maintained Erdős Problems database at commit 8138974387d9030542daabe67faaa33eff9356f8 lists Problem 50 as open. The unresolved remainder is the full displayed statement. Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 50: Let $\varphi$ denote Euler's totient function. A function $f : \mathbb{R} \to \mathbb{R}$ is called the asymptotic distribution function of $\varphi(n)/n$ if for every $c \in [0, 1]$, the set $\{n \in \mathbb{N} : \varphi(n) < cn\}$ has a natural density, and this density equals $f(c)$. Determine whether the following holds: for every function $f : \mathbb{R} \to \mathbb{R}$ that is the asymptotic distribution function of $\varphi(n)/n$, there do not exist $x \in [0, 1]$ and $y > 0$ such that $f$ has derivative $y$ at $x$ within the interval $[0, 1]$.^[1]

1Packet records

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Record	Kind	Assessment
OPEN as of 2026-07-31. The maintained Erdős Problems database at commit 8138974387d9030542daabe67faaa33eff9356f8 lists Problem 50 as open. The unresolved remainder is the full displayed statement. Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 50: Let $\varphi$ denote Euler's totient function. A function $f : \mathbb{R} \to \mathbb{R}$ is called the asymptotic distribution function of $\varphi(n)/n$ if for every $c \in [0, 1]$, the set $\{n \in \mathbb{N} : \varphi(n) < cn\}$ has a natural density, and this density equals $f(c)$. Determine whether the following holds: for every function $f : \mathbb{R} \to \mathbb{R}$ that is the asymptotic distribution function of $\varphi(n)/n$, there do not exist $x \in [0, 1]$ and $y > 0$ such that $f$ has derivative $y$ at $x$ within the interval $[0, 1]$.	Result	Evidence: Supported Backed by a cited source or by evidence short of a proof.
claim · Claim 1 OPEN as of 2026-07-31. The maintained Erdős Problems database at commit 8138974387d9030542daabe67faaa33eff9356f8 lists Problem 50 as open. The unresolved remainder is the full displayed statement. Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 50: Let $\varphi$ denote Euler's totient function. A function $f : \mathbb{R} \to \mathbb{R}$ is called the asymptotic distribution function of $\varphi(n)/n$ if for every $c \in [0, 1]$, the set $\{n \in \mathbb{N} : \varphi(n) < cn\}$ has a natural density, and this density equals $f(c)$. Determine whether the following holds: for every function $f : \mathbb{R} \to \mathbb{R}$ that is the asymptotic distribution function of $\varphi(n)/n$, there do not exist $x \in [0, 1]$ and $y > 0$ such that $f$ has derivative $y$ at $x$ within the interval $[0, 1]$.^[1] Relevance to this problem For erdos problem 50, pins the dated research frontier: OPEN as of 2026-07-31. The maintained Erdős Problems database at commit 8138974387d9030542daabe67faaa33eff9356f8 lists Problem 50 as open. The unresolved remainder is the full displayed statement. Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 50: Let. Evidence SupportedBacked by a cited source or by evidence short of a proof. Record state reported Scope No narrower scope recorded Sources Reference 1 Details OPEN as of 2026-07-31. The maintained Erdős Problems database at commit 8138974387d9030542daabe67faaa33eff9356f8 lists Problem 50 as open. The unresolved remainder is the full displayed statement. A complete resolution must satisfy this condition: Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 50: Let $\varphi$ denote Euler's totient function. A function $f : \mathbb{R} \to \mathbb{R}$ is called the asymptotic distribution function of $\varphi(n)/n$ if for every $c \in [0, 1]$, the set $\{n \in \mathbb{N} : \varphi(n) < cn\}$ has a natural density, and this density equals $f(c)$. Determine whether the following holds: for every function $f : \mathbb{R} \to \mathbb{R}$ that is the asymptotic distribution function of $\varphi(n)/n$, there do not exist $x \in [0, 1]$ and $y > 0$ such that $f$ has derivative $y$ at $x$ within the interval $[0, 1]$. Open full record →
Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 50: Let $\varphi$ denote Euler's totient function. A function $f : \mathbb{R} \to \mathbb{R}$ is called the asymptotic distribution function of $\varphi(n)/n$ if for every $c \in [0, 1]$, the set $\{n \in \mathbb{N} : \varphi(n) < cn\}$ has a natural density, and this density equals $f(c)$. Determine whether the following holds: for every function $f : \mathbb{R} \to \mathbb{R}$ that is the asymptotic distribution function of $\varphi(n)/n$, there do not exist $x \in [0, 1]$ and $y > 0$ such that $f$ has derivative $y$ at $x$ within the interval $[0, 1]$.	Trace	Outcome: Open Evidence: Reported Stated by one agent or source, not independently checked.
attempt · Route 1 Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 50: Let $\varphi$ denote Euler's totient function. A function $f : \mathbb{R} \to \mathbb{R}$ is called the asymptotic distribution function of $\varphi(n)/n$ if for every $c \in [0, 1]$, the set $\{n \in \mathbb{N} : \varphi(n) < cn\}$ has a natural density, and this density equals $f(c)$. Determine whether the following holds: for every function $f : \mathbb{R} \to \mathbb{R}$ that is the asymptotic distribution function of $\varphi(n)/n$, there do not exist $x \in [0, 1]$ and $y > 0$ such that $f$ has derivative $y$ at $x$ within the interval $[0, 1]$.^[1] Relevance to this problem For Non-existence of positive derivatives for the distribution function of Euler's totient ratio, record erdos-problem-50-attempt-resolution-route (“Resolve the stated acceptance condition”) documents a concrete method, search boundary, or failed route. The record states: Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 50: Let $\varphi$ denote Euler's totient function. Evidence ReportedStated by one agent or source, not independently checked. Record state open strategy Scope No narrower scope recorded Sources Reference 1 What happened Target the displayed statement directly. Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 50: Let $\varphi$ denote Euler's totient function. A function $f : \mathbb{R} \to \mathbb{R}$ is called the asymptotic distribution function of $\varphi(n)/n$ if for every $c \in [0, 1]$, the set $\{n \in \mathbb{N} : \varphi(n) < cn\}$ has a natural density, and this density equals $f(c)$. Determine whether the following holds: for every function $f : \mathbb{R} \to \mathbb{R}$ that is the asymptotic distribution function of $\varphi(n)/n$, there do not exist $x \in [0, 1]$ and $y > 0$ such that $f$ has derivative $y$ at $x$ within the interval $[0, 1]$. Preserve exact hypotheses, source locators, and any finite certificates so later work can distinguish a full resolution from partial progress. Related: Claim 1 Open full record →
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Notes and companion material

Original intake status. OPEN as of 2026-07-31. The maintained Erdős Problems database at commit 8138974387d9030542daabe67faaa33eff9356f8 lists Problem 50 as open. The unresolved remainder is the full displayed statement.

The maintained database entry for Erdős Problem 50 was open at commit 8138974387d9030542daabe67faaa33eff9356f8.
The pinned Formal Conjectures declaration was matched by problem number and source locator.
The controlled TheoremDB source corpus was checked for an already published record with the same slug.

How the 2 records connect

The overview places each record once. The relation list includes shared dependencies and names both ends of each link.

ProblemNon-existence of positive derivatives for the distribution function of Euler's totient ratio

All 1 recorded relations between these records and the problem

Resolve the stated acceptance condition addresses Current status and unresolved remainder

2See also

Merging of orbits under adding the product of nonzero digitsnumber theory
Sharp multipliers for balanced binary productsnumber theory
Prime exceptions to connectivity of the Markoff graphnumber theory

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Plain text

“Non-existence of positive derivatives for the distribution function of Euler's totient ratio.” TheoremDB. P2998. Problem statement; statement text SHA-256 cb3bd74d58a7b6dc991ba3550c785b25ee74fcd56ff75c59adbd09a71f01b3fe. https://theoremdb.org/statement/?ref=P2998

BibTeX

@misc{theoremdb-problem-cb3bd74d58a7b6dc991ba3550c785b25ee74fcd56ff75c59adbd09a71f01b3fe,
  title = {{Non-existence of positive derivatives for the distribution function of Euler's totient ratio}},
  howpublished = {TheoremDB},
  note = {Problem statement; statement text SHA-256 cb3bd74d58a7b6dc991ba3550c785b25ee74fcd56ff75c59adbd09a71f01b3fe},
  url = {https://theoremdb.org/statement/?ref=P2998}
}

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This problem includes 2 records joined by 1 typed links, sourced from erdosproblems.com^[1], current as of July 31, 2026.

1References

Packet source. Erdős Problems database, Problem 50, maintained status record. Erdős Problems record 50, checked 2026-08-01. Problem 50; status field and linked bibliography. ↗reference database · reference source · commit 8138974387d9030542daabe67faaa33eff9356f8 · checked 2026-07-31Source use: original summary.Records the current open status and links the literature attached to this exact numbered problem.Also cited at Problem 50; status snapshot 8138974387d9030542daabe67faaa33eff9356f8.Also cited at See dataset.references[0] for the exact external source and locator.Also cited at Editorial research route recorded 2026-07-31.Source used to formulate or check the problem record.Source used to assess the problem's recorded status.For Non-existence of positive derivatives for the distribution function of Euler's totient ratio: Records the current open status and links the literature attached to this exact numbered problem.Source named by the research packet.
Erdős problem database, data/problems.yaml, commit 8138974387d9030542daabe67faaa33eff9356f8. Problem 50 entry in data/problems.yaml. ↗reference database · reference source · commit 8138974387d9030542daabe67faaa33eff9356f8 · checked 2026-07-31Source use: original summary.Pins the maintained database snapshot used for this release's dated status decision.Source used to assess the problem's recorded status.For Non-existence of positive derivatives for the distribution function of Euler's totient ratio: Pins the maintained database snapshot used for this release's dated status decision.
Google DeepMind, Formal Conjectures, formal statement for Erdős Problem 50. GitHub commit bb1c69dab14ad895ac1ccdb7b0c0e0a67c8eb3e1. FormalConjectures/ErdosProblems/50.lean:L75; theorem erdos_50; commit bb1c69dab14ad895ac1ccdb7b0c0e0a67c8eb3e1. ↗reference database · primary source · commit bb1c69dab14ad895ac1ccdb7b0c0e0a67c8eb3e1 · checked 2026-07-31Source use: original summary.Supplies the pinned formal declaration whose human-readable editorial statement is published here.Source used to assess the problem's recorded status.For Non-existence of positive derivatives for the distribution function of Euler's totient ratio: Supplies the pinned formal declaration whose human-readable editorial statement is published here.

Original TheoremDB editorial prose based on a pinned formal declaration and the maintained status database.

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[#P2998] Non-existence of positive derivatives for the distribution function of Euler's totient ratio

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