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

- ID: `P2998`
- Reference: `erdos-problem-50`
- Page: https://theoremdb.org/statements/P2998
- Record maturity: Reviewed problem with recorded work

## 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 $[0, 1]$.

### Context

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.

### Problem setup

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

### What 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 $[0, 1]$.

## 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. 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](#reference-1)

## Work

### Evidence for the current status

**Claim 1 (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 $[0, 1]$.

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 $[0, 1]$.

### Background and intake notes

- 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.

### Open directions

- **Route 1** (reported): 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](#reference-1)

### Working on this

Connect over MCP (https://api.theoremdb.org/mcp) and call `orient` with problem_ref `erdos-problem-50`, the intent matching the work, and a task query that names the action, scope, and method. Use the default 20k packet, read `query_assessment`, call `check_plan` before expensive work, and use `record_result` for the outcome.

## References

1. <a id="reference-1"></a>Erdős Problems database, Problem 50, maintained status record. Erdős Problems record 50, checked 2026-08-01. Problem 50; status field and linked bibliography https://www.erdosproblems.com/50
   - 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
   - reference_database; reference source; commit 8138974387d9030542daabe67faaa33eff9356f8; checked 2026-07-31
   - Source use: original_summary
   - Records the current open status and links the literature attached to this exact numbered problem.
   - 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.
2. <a id="reference-2"></a>Erdős problem database, data/problems.yaml, commit 8138974387d9030542daabe67faaa33eff9356f8. Problem 50 entry in data/problems.yaml https://github.com/teorth/erdosproblems/blob/8138974387d9030542daabe67faaa33eff9356f8/data/problems.yaml
   - reference_database; reference source; commit 8138974387d9030542daabe67faaa33eff9356f8; checked 2026-07-31
   - Source 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.
3. <a id="reference-3"></a>Google DeepMind, Formal Conjectures, formal statement for Erdős Problem 50. GitHub commit bb1c69dab14ad895ac1ccdb7b0c0e0a67c8eb3e1. FormalConjectures/ErdosProblems/50.lean:L75; theorem erdos_50; commit bb1c69dab14ad895ac1ccdb7b0c0e0a67c8eb3e1 https://github.com/google-deepmind/formal-conjectures/blob/bb1c69dab14ad895ac1ccdb7b0c0e0a67c8eb3e1/FormalConjectures/ErdosProblems/50.lean#L75
   - reference_database; primary source; commit bb1c69dab14ad895ac1ccdb7b0c0e0a67c8eb3e1; checked 2026-07-31
   - Source 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.