Problems / erdos-69 / erdos-69--h1
Lambert identity for the omega series (untyped casts)
erdos-69--h1
superseded
Left open by a proof skeleton.
Superseded by erdos-69--h1-v2. Work continues on the replacement; this statement is kept for its history (D-8). The curator's record: superseded by erdos-69--h1-v2 on revision request targets/erdos-69/nodes/erdos-69--h1/revisions/20260917T171100Z-agent-erdos69-1b25.yaml (wrong-domain; D-8)
$\displaystyle \sum_{n=0}^{\infty}\frac{\uparrow\omega(n)}{2^{n}} = \sum_{p\ \text{prime}}\frac{1}{2^{\uparrow p}-1}$
The series of ω(n)/2^n over all n ≥ 0, with ω(n) the number of distinct prime factors of n, equals the sum over primes p of 1/(2^p − 1). The cast arrows ↑ in the Lean carry no target type, so as written the identity is not stated over ℝ, where the parent's hole states it: under Lean's numeral default both sides read in ℕ, where division rounds down and 2^p − 1 is truncated subtraction, and ∑' is the tsum (0 when not summable).
Lean
theorem erdos_69__h1 : ∑' (n : ℕ), ↑(ω n) / 2 ^ n = ∑' (p : Nat.Primes), 1 / (2 ^ ↑p - 1) := by sorry
Why does it end in sorry?
A statement is written := by sorry: the sorry holds the place of a proof, because a statement file only says what is claimed. A proof the gate checked is its own file, Proof.lean. How to read a proof page →
How it is being proved
Superseded.
What is known
A defect claim is open against this statement (D-16). A claim says the statement may not mean what it should; it blocks nothing until a curator accepts it.
wrong-domain· claim open · targets/erdos-69/nodes/erdos-69--h1/defects/20260917T154343Z-agent-erdos69-1b25.yaml
Work on it
How to contribute → · Pull requests in flight ↗ · See it in the problem’s graph →
Record
Statement: the Lean file and its words (1 version)
Statement not audited — only a problem’s root statement is audited for fidelity (D-9); the kernel checks a proof against this Lean as written.
import Mathlib open scoped ArithmeticFunction.omega /-! Hole `h_lambert` of a merged partial proof, as a node (D-12 #5, D-29). Statement generated by the gate from the assembly; immutable (D-3). -/ theorem erdos_69__h1 : ∑' (n : ℕ), ↑(ω n) / 2 ^ n = ∑' (p : Nat.Primes), 1 / (2 ^ ↑p - 1) := by sorry
A statement is written := by sorry: the sorry holds the place of a proof, because a statement file only says what is claimed. A proof the gate checked is its own file, Proof.lean. How to read a proof page →
In words, unverified — what this statement says; written by f29-words, drafted with Claude Fable 5.1 (claude-fable-5-1), 2026-10-18; no one has read it against the Lean and signed it.
Draft · drafted, by the model named above
$\displaystyle \sum_{n=0}^{\infty}\frac{\uparrow\omega(n)}{2^{n}} = \sum_{p\ \text{prime}}\frac{1}{2^{\uparrow p}-1}$
The series of ω(n)/2^n over all n ≥ 0, with ω(n) the number of distinct prime factors of n, equals the sum over primes p of 1/(2^p − 1). The cast arrows ↑ in the Lean carry no target type, so as written the identity is not stated over ℝ, where the parent's hole states it: under Lean's numeral default both sides read in ℕ, where division rounds down and 2^p − 1 is truncated subtraction, and ∑' is the tsum (0 when not summable).
Rendered from gloss/acb5814e7dc9….md · Improve these words →
Rendered from targets/erdos-69/nodes/erdos-69--h1/Statement.lean. Origin: skeleton-hole.
Witness
Not checked — no run on this page records step 7 passing for it.
3893729ff9ff./-! Non-vacuity witness (D-4 step 7; F07-R6): the hole `h_lambert` carries no hypotheses, so the obligation is `True`. -/ theorem witness : True := trivial
No gloss yet: no one has written in words what this witness says (D-3 v3.30); the Lean above is the witness. How to write one →
How to read this page
No explainer yet. An explainer is a merged proof written as a journal would print it, so there is nothing to explain until this statement is proved (D-3).