Problems / erdos-1094 / erdos-1094--h3--h3--h1--h1--h1--h2
-ζ'/ζ minus the pole extends to Re s ≥ 1
erdos-1094--h3--h3--h1--h1--h1--h2
open
Hole h_ext of erdos-1094--h3--h3--h1--h1--h1 (split by 20261018T200210Z-t1018-e-partial.lean).
$$\sum_n \Lambda(n) n^{-s} - \frac{1}{s-1}$ extends continuously to $\Re s\ge 1$$
From ζ's simple pole of residue 1 at s = 1 and ζ(s) ≠ 0 on Re s ≥ 1 (Mathlib: riemannZeta_residue_one, riemannZeta_ne_zero_of_one_le_re, ArithmeticFunction.LSeries_vonMangoldt_eq_deriv_riemannZeta_div).
Lean
theorem erdos_1094__h3__h3__h1__h1__h1__h2 : ∃ (G : ℂ → ℂ),
ContinuousOn G {s : ℂ | (1 : ℝ) ≤ s.re} ∧
∀ (s : ℂ),
(1 : ℝ) < s.re →
G s = LSeries (fun (n : ℕ) => (↑((ArithmeticFunction.vonMangoldt : ℕ → ℝ) n) : ℂ)) s - (1 : ℂ) / (s - (1 : ℂ)) := by
sorryWhy 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
Open. A direct proof closes it.
What is known
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
import Nodes.«erdos-1094--h3--h3--h1--h1--h1--h2».Context
open scoped Nat
/-! Hole `h_ext` 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_1094__h3__h3__h1__h1__h1__h2 : ∃ (G : ℂ → ℂ),
ContinuousOn G {s : ℂ | (1 : ℝ) ≤ s.re} ∧
∀ (s : ℂ),
(1 : ℝ) < s.re →
G s = LSeries (fun (n : ℕ) => (↑((ArithmeticFunction.vonMangoldt : ℕ → ℝ) n) : ℂ)) s - (1 : ℂ) / (s - (1 : ℂ)) := by
sorryA 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 t1018-e, drafted with claude-opus-5-5, 2026-10-18; no one has read it against the Lean and signed it.
Draft · drafted, by the model named above
$$\sum_n \Lambda(n) n^{-s} - \frac{1}{s-1}$ extends continuously to $\Re s\ge 1$$
From ζ's simple pole of residue 1 at s = 1 and ζ(s) ≠ 0 on Re s ≥ 1 (Mathlib: riemannZeta_residue_one, riemannZeta_ne_zero_of_one_le_re, ArithmeticFunction.LSeries_vonMangoldt_eq_deriv_riemannZeta_div).
Rendered from gloss/61f088846ab1….md · Improve these words →
Rendered from targets/erdos-1094/nodes/erdos-1094--h3--h3--h1--h1--h1--h2/Statement.lean. Origin: compiler-derived.
Witness
Not checked — no run on this page records step 7 passing for it.
d610aa356d48.import Mathlib open scoped Nat 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).