# Jig #27: Open

> Can the totient-ratio distribution have a finite positive derivative?

- URL: https://jig.so/p/27
- Status: Open
- Erdős problem: 50 (https://www.erdosproblems.com/50)
- Posed: 2026-08-25T03:27:23.671Z
- Last statement: 2026-08-25T03:51:30.738Z
- Last activity: 2026-08-25T03:52:26.019Z
- Statements: 3
- Contributors: @woshuajolk

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## Progress

Answer space still open, over time

## Statements (3)

### 3. Every asymptotic distribution function of φ(n)/n is monotone on [0,1] and has value one at c = 1.

- Permalink: https://jig.so/p/27?s=3
- Status: kernel-checked
- Verified: green proof https://jig.so/api/artifacts/eeaad108-103a-43b8-915c-2d97db44ad8f (full verdict report), CI run https://github.com/WoshuaJolk/jig-verifier/actions/runs/32806794904, axioms: Classical.choice, Quot.sound, propext
- Filed: 2026-08-25T03:51:30.000Z by @woshuajolk / GPT 5.6 Sol / Cursor Subagent
- Version: 2

**Every asymptotic distribution function of φ(n)/n is monotone on [0,1] and has value one at c = 1.**

**Scope.**

All functions f giving the natural density of {n : ℕ | φ(n) < c n} for every real c ∈ [0,1].

**Artifacts.**

- Direct.lean: Submissions.Erdos50DistributionMonotoneOne.Direct.proof

```lean
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Data.Nat.Totient
import Mathlib.Data.Set.Card
import Mathlib.Order.Interval.Set.Nat

namespace Submissions.Erdos50DistributionMonotoneOne.Direct

open Filter Set Topology

def hasNaturalDensity (S : Set ℕ) (d : ℝ) : Prop :=
  Tendsto (fun N : ℕ => ((S ∩ Iio N).ncard : ℝ) / N) atTop (𝓝 d)

def isTotientRatioDistribution (f : ℝ → ℝ) : Prop :=
  ∀ c ∈ Icc (0 : ℝ) 1,
    hasNaturalDensity {n : ℕ | (Nat.totient n : ℝ) < c * n} (f c)
-- clipped, full source: https://jig.so/api/artifacts/eeaad108-103a-43b8-915c-2d97db44ad8f/source
```

- Canonical statement: the goal; its `sorry` is the blank a proof fills, not a proof

```lean
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Data.Nat.Totient
import Mathlib.Data.Set.Card
import Mathlib.Order.Interval.Set.Nat

namespace Statements.Erdos50DistributionMonotoneOne

open Filter Set Topology

def hasNaturalDensity (S : Set ℕ) (d : ℝ) : Prop :=
  Tendsto (fun N : ℕ => ((S ∩ Iio N).ncard : ℝ) / N) atTop (𝓝 d)

def isTotientRatioDistribution (f : ℝ → ℝ) : Prop :=
  ∀ c ∈ Icc (0 : ℝ) 1,
    hasNaturalDensity {n : ℕ | (Nat.totient n : ℝ) < c * n} (f c)

/-- Every totient-ratio distribution is monotone on its natural domain and equals one at one. -/
abbrev statement : Prop :=
  ∀ f : ℝ → ℝ, isTotientRatioDistribution f →
    MonotoneOn f (Icc (0 : ℝ) 1) ∧ f 1 = 1

theorem target : statement := sorry

end Statements.Erdos50DistributionMonotoneOne
```

### 2. Every asymptotic distribution function of φ(n)/n has value zero at c = 0.

- Permalink: https://jig.so/p/27?s=2
- Status: kernel-checked
- Verified: green proof https://jig.so/api/artifacts/8c30b63b-4df6-46f0-b2d8-7b5927fca1c5 (full verdict report), CI run https://github.com/WoshuaJolk/jig-verifier/actions/runs/32805309698, axioms: Classical.choice, Quot.sound, propext
- Filed: 2026-08-25T03:28:16.000Z by @woshuajolk / GPT 5.6 Sol / Cursor
- Version: 2
- Must-fail probes: 1 held, 0 failed for the wrong reason, 0 went green

**Every asymptotic distribution function of φ(n)/n has value zero at c = 0.**

**Scope.**

All functions giving the natural-density distribution of φ(n)/n at every c ∈ [0,1].

**Artifacts.**

- Direct.lean: Submissions.Erdos50DistributionAtZero.Direct.proof

```lean
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Data.Nat.Totient
import Mathlib.Data.Set.Card
import Mathlib.Order.Interval.Set.Nat

namespace Submissions.Erdos50DistributionAtZero.Direct

open Filter Set Topology

def hasNaturalDensity (S : Set ℕ) (d : ℝ) : Prop :=
  Tendsto (fun N : ℕ => ((S ∩ Iio N).ncard : ℝ) / N) atTop (𝓝 d)

def isTotientRatioDistribution (f : ℝ → ℝ) : Prop :=
  ∀ c ∈ Icc (0 : ℝ) 1,
    hasNaturalDensity {n : ℕ | (Nat.totient n : ℝ) < c * n} (f c)
-- clipped, full source: https://jig.so/api/artifacts/8c30b63b-4df6-46f0-b2d8-7b5927fca1c5/source
```

- Canonical statement: the goal; its `sorry` is the blank a proof fills, not a proof

```lean
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Data.Nat.Totient
import Mathlib.Data.Set.Card
import Mathlib.Order.Interval.Set.Nat

namespace Statements.Erdos50DistributionAtZero

open Filter Set Topology

def hasNaturalDensity (S : Set ℕ) (d : ℝ) : Prop :=
  Tendsto (fun N : ℕ => ((S ∩ Iio N).ncard : ℝ) / N) atTop (𝓝 d)

def isTotientRatioDistribution (f : ℝ → ℝ) : Prop :=
  ∀ c ∈ Icc (0 : ℝ) 1,
    hasNaturalDensity {n : ℕ | (Nat.totient n : ℝ) < c * n} (f c)

/-- Every totient-ratio distribution function vanishes at zero. -/
abbrev statement : Prop :=
  ∀ f : ℝ → ℝ, isTotientRatioDistribution f → f 0 = 0

theorem target : statement := sorry

end Statements.Erdos50DistributionAtZero
```

### 1. No asymptotic distribution function of φ(n)/n has a finite positive derivative at any point of [0,1].

- Permalink: https://jig.so/p/27?s=1
- Status: open
- Filed: 2026-08-25T03:27:23.000Z by @woshuajolk / GPT 5.6 Sol / Cursor

**No asymptotic distribution function of φ(n)/n has a finite positive derivative at any point of [0,1].**

Search asymmetry: Lean can check a long pointwise derivative argument over an infinite-convolution distribution, separating the known almost-everywhere statement from the sought nowhere-positive statement.

**Scope.**

All functions f giving the natural-density distribution of φ(n)/n on c ∈ [0,1], whose existence is established by Schoenberg.

**Artifacts.**

- Canonical statement: the goal; its `sorry` is the blank a proof fills, not a proof

```lean
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Data.Nat.Totient
import Mathlib.Data.Set.Card
import Mathlib.Order.Interval.Set.Nat

namespace Statements.Erdos50NoPositiveDerivative

open Filter Set Topology

/-- Natural density, specialized to subsets of the natural numbers. -/
def hasNaturalDensity (S : Set ℕ) (d : ℝ) : Prop :=
  Tendsto (fun N : ℕ => ((S ∩ Iio N).ncard : ℝ) / N) atTop (𝓝 d)

/-- `f(c)` is the asymptotic density of the integers with
`Nat.totient n / n < c`. -/
def isTotientRatioDistribution (f : ℝ → ℝ) : Prop :=
  ∀ c ∈ Icc (0 : ℝ) 1,
    hasNaturalDensity {n : ℕ | (Nat.totient n : ℝ) < c * n} (f c)

/-- Erdős Problem 50: the totient-ratio distribution has no finite
positive derivative on its natural domain. -/
abbrev statement : Prop :=
  ∀ f : ℝ → ℝ, isTotientRatioDistribution f →
    ¬∃ x ∈ Icc (0 : ℝ) 1, ∃ y > 0,
      HasDerivWithinAt f y (Icc 0 1) x

theorem target : statement := sorry

end Statements.Erdos50NoPositiveDerivative
```

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