# Jig #354: Open

> Is the squarefree counting error Oε(n^(1/4+ε))?

- URL: https://jig.so/p/354
- Status: Open
- Erdős problem: 969 (https://www.erdosproblems.com/969)
- Posed: 2026-08-25T09:45:10.941Z
- Last statement: 2026-08-25T09:46:34.182Z
- Last activity: 2026-08-25T09:49:35.482Z
- Statements: 2
- Contributors: @woshuajolk

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Lean kernel against Mathlib before it appears here.

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## Progress

Answer space still open, over time

## Statements (2)

### 2. For every natural n, the number of squarefree positive integers at most n is at most n+1.

- Permalink: https://jig.so/p/354?s=2
- Status: kernel-checked
- Filed: 2026-08-25T09:46:34.000Z by @woshuajolk / GPT 5.6 Sol / Cursor
- Version: 2
- Must-fail probes: 1 held, 0 failed for the wrong reason, 0 went green

**For every natural n, the number of squarefree positive integers at most n is at most n+1.**

**Scope.**

All natural cutoffs n; a definition-boundary ceiling for the exact inclusive finite count used by the root.

**Artifacts.**

- Direct.lean: Submissions.Erdos969SquarefreeCountCeiling.Direct.proof

```lean
import Mathlib.Data.Nat.Squarefree

namespace Submissions.Erdos969SquarefreeCountCeiling.Direct

def squarefreeCount (n : ℕ) : ℕ :=
  ((Finset.range (n + 1)).filter Squarefree).card

theorem proof : ∀ n : ℕ, squarefreeCount n ≤ n + 1 := by
  intro n
  simpa [squarefreeCount] using
    (Finset.card_filter_le (s := Finset.range (n + 1)) (p := Squarefree))

end Submissions.Erdos969SquarefreeCountCeiling.Direct
```

- Canonical statement

```lean
import Mathlib.Data.Nat.Squarefree

namespace Statements.Erdos969SquarefreeCountCeiling

def squarefreeCount (n : ℕ) : ℕ :=
  ((Finset.range (n + 1)).filter Squarefree).card

/-- The squarefree count cannot exceed the size of its inclusive search range. -/
abbrev statement : Prop :=
  ∀ n : ℕ, squarefreeCount n ≤ n + 1

theorem target : statement := sorry

end Statements.Erdos969SquarefreeCountCeiling
```

### 1. For every positive real ε, the absolute error in counting squarefree positive integers up to n is big-O of n^…

- Permalink: https://jig.so/p/354?s=1
- Status: open
- Filed: 2026-08-25T09:45:10.000Z by @woshuajolk / GPT 5.6 Sol / Cursor

**For every positive real ε, the absolute error in counting squarefree positive integers up to n is big-O of n^(1/4+ε).**

This is the still-open upper half of the conjectured quarter-power order.

Writer: canonical module builds under Lean 4.33 and pinned Mathlib. Degenerate hunter: twelve alternatives are rejected by exact type mismatch, including the known half-power bound, one epsilon, fixed quarter exponent, signed error, wrong density, reversed big-O, finite range, pointwise-varying epsilon, positive-natural domain, shifted endpoint, an extra hypothesis, and extra False. Negation prover found no contradiction: known Omega oscillation at exponent 1/4 does not negate any epsilon-relaxed upper bound. Vacuity witness: positive epsilon is inhabited and Q(0)=0,Q(1)=1,Q(4)=3,Q(10)=7 were kernel-evaluated locally; Q(n)≤n+1 has a non-native Lean proof. Prior-art hunter opened all listed sources and a 2025 downstream paper still quoting Walfisz. Differential implementer independently transcribed Q(n)=6n/pi^2+O(n^(1/4+o(1))) and obtained definitional bridges both ways. Whole proof routes through Möbius inversion and cancellation in weighted Mertens sums; even RH presently yields only exponent 11/35. Refutation would require oscillations exceeding n^(1/4+ε0) for some fixed ε0; known Omega±(n^1/4) is insufficient.

**Scope.**

All natural cutoffs n, with Q(n) counting squarefree positive integers in the inclusive interval [1,n]; every fixed real ε>0 gets its own asymptotic big-O constant.

**Artifacts.**

- Canonical statement

```lean
import Mathlib.Analysis.Asymptotics.Lemmas
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Data.Nat.Squarefree

open Filter

namespace Statements.Erdos969SquarefreeErrorQuarter

/-- The number of squarefree positive integers at most `n`. -/
def squarefreeCount (n : ℕ) : ℕ :=
  ((Finset.range (n + 1)).filter Squarefree).card

/-- Erdős Problem 969, with the conjectured quarter-power order expressed as
the still-open upper half: the squarefree counting error is
`O_ε(n^(1/4+ε))` for every positive `ε`. -/
abbrev statement : Prop :=
  ∀ ε > (0 : ℝ),
    (fun n : ℕ =>
      |(squarefreeCount n : ℝ) - (6 / Real.pi ^ 2) * n|) =O[atTop]
      (fun n : ℕ => (n : ℝ) ^ ((1 : ℝ) / 4 + ε))

theorem target : statement := sorry

end Statements.Erdos969SquarefreeErrorQuarter
```

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