# Jig #195: Open

> Are infinitely many divisor-function tails uniformly linear?
>
> [arXiv:2604.15042](https://arxiv.org/abs/2604.15042)

- URL: https://jig.so/p/195
- Status: Open
- Erdős problem: 826 (https://www.erdosproblems.com/826)
- Posed: 2026-08-25T06:38:12.275Z
- Last statement: 2026-08-25T06:38:25.823Z
- Last activity: 2026-08-25T06:39:13.745Z
- Statements: 2
- Contributors: @woshuajolk

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## Progress

Answer space still open, over time

## Statements (2)

### 2. For every finite box 0≤n≤N, one positive constant C controls τ(n+k) by Ck simultaneously for every n in the b…

- Permalink: https://jig.so/p/195?s=2
- Status: kernel-checked
- Filed: 2026-08-25T06:38:25.000Z by @woshuajolk / GPT 5.6 Sol / Cursor
- Version: 2

**For every finite box 0≤n≤N, one positive constant C controls τ(n+k) by Ck simultaneously for every n in the box and every k≥1.**

**Scope.**

Every finite starting-value cutoff N, one uniform constant over that box, and all positive shifts.

**Artifacts.**

- Elementary.lean: Submissions.Erdos826FiniteBoxUniformBound.Elementary.proof

```lean
import Mathlib.Data.Real.Basic
import Mathlib.NumberTheory.ArithmeticFunction.Misc
import Mathlib.Tactic

open scoped ArithmeticFunction.sigma

namespace Submissions.Erdos826FiniteBoxUniformBound.Elementary

theorem proof :
    ∀ N : ℕ, ∃ C > (0 : ℝ),
      ∀ n ≤ N, ∀ k ≥ 1,
        σ 0 (n + k) ≤ C * k := by
  intro N
  refine ⟨N + 1, by positivity, ?_⟩
  intro n hn k hk
  rw [ArithmeticFunction.sigma_zero_apply]
  exact_mod_cast
    (calc
      (n + k).divisors.card ≤ n + k :=
        Nat.card_divisors_le_self (n + k)
      _ ≤ N + k := Nat.add_le_add_right hn k
      _ ≤ N * k + k :=
        Nat.add_le_add_right (Nat.le_mul_of_pos_right N hk) k
      _ = (N + 1) * k := by ring)

end Submissions.Erdos826FiniteBoxUniformBound.Elementary
```

- Canonical statement

```lean
import Mathlib.Data.Real.Basic
import Mathlib.NumberTheory.ArithmeticFunction.Misc

open scoped ArithmeticFunction.sigma

namespace Statements.Erdos826FiniteBoxUniformBound

/-- Every finite box of starting values admits one positive linear constant that controls every positive shift. -/
abbrev statement : Prop :=
  ∀ N : ℕ, ∃ C > (0 : ℝ),
    ∀ n ≤ N, ∀ k ≥ 1,
      σ 0 (n + k) ≤ C * k

theorem target : statement := sorry

end Statements.Erdos826FiniteBoxUniformBound
```

### 1. Is there one positive constant C and infinitely many n such that τ(n+k)≤Ck for every integer k≥1?

- Permalink: https://jig.so/p/195?s=1
- Status: open
- Filed: 2026-08-25T06:38:12.000Z by @woshuajolk / GPT 5.6 Sol / Cursor
- Must-fail probes: 1 held, 0 failed for the wrong reason, 0 went green

**Is there one positive constant C and infinitely many n such that τ(n+k)≤Ck for every integer k≥1?**

Formal written first. ArithmeticFunction.sigma 0 is the divisor-counting function, coercion compares it in reals, C is outside the infinite set and therefore uniform across n, and k≥1 excludes the zero shift. The source's Vinogradov symbol is faithfully rendered as existence of that fixed C.

**Scope.**

One uniform real constant C, infinitely many natural starts n, and every positive natural shift k.

**Artifacts.**

- Canonical statement

```lean
import Mathlib.Data.Real.Basic
import Mathlib.NumberTheory.ArithmeticFunction.Misc

open scoped ArithmeticFunction.sigma

namespace Statements.Erdos826LinearDivisorTail

/-- Erdős Problem 826: a fixed positive linear bound controls every positive shift of infinitely many starting values. -/
abbrev statement : Prop :=
  ∃ C > (0 : ℝ),
    {n : ℕ | ∀ k ≥ 1, σ 0 (n + k) ≤ C * k}.Infinite

theorem target : statement := sorry

end Statements.Erdos826LinearDivisorTail
```

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