LemmaStatement: Literature-sourcedProof: AI-generatedPipeline-generatedprecheck passverified 2026-09-24 (gpt-6-sol)

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

A left-half-plane bound for the logarithmic derivative of zeta

Statement

Assume countable choice.

Fix $η > 0$ . If $ℜ s \leq - 1$ and $s$ stays at distance at least $η$ from every negative even integer, then $ζ^{'} (s) / ζ (s) = O_{η} (\log (∣ s ∣ + 2))$ .

Proof

Given: Countable choice and the displayed distance condition in the left half-plane.

1.1givenalgebra

Take logarithmic derivatives of The Riemann zeta function satisfies the classical sine-gamma functional equation. The logarithmic derivative of the sine factor is $\frac{π}{2} \cot (π s / 2)$ , which is uniformly bounded under the distance condition. Since $Re (1 - s) \geq 2$ , the absolutely convergent series for $- ζ^{'} / ζ$ in The logarithmic derivative of the zeta Dirichlet series is the Dirichlet series of the von Mangoldt function on Re s greater than 1 bounds $ζ^{'} (1 - s) / ζ (1 - s)$ uniformly.

2.1step 1.1algebra∎

Applying Stirling's formula for Gamma in the right half-plane to $1 - s$ and differentiating its uniform analytic remainder by Cauchy's estimate on smaller sectorial disks gives $Γ^{'} (1 - s) / Γ (1 - s) = O (\log (∣ s ∣ + 2))$ for $Re (1 - s) \geq 2$ . Combining the finitely many factor bounds proves the assertion.

Depends on

Used by

A smoothed von Mangoldt explicit formula Theorem
The truncated von Mangoldt explicit formula Theorem

Dependency tree · two levels

19 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Kiran S. Kedlaya, Analytic Number Theory, §10.3 (standard reference, not scraped)