Alphabeta Math
LemmaStatement: Literature-sourcedProof: AI-generatedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-07
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A left-half-plane bound for the logarithmic derivative of zeta

Statement

Fix η>0. If ℜs≤−1 and s stays at distance at least η from every negative even integer, then ζ′(s)/ζ(s)=Oη(log⁡(∣s∣+2)).

Proof

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

1.1

Take logarithmic derivatives of the zeta functional equation. The logarithmic derivative of the sine factor is π2cot⁡(πs/2), which is uniformly bounded under the distance condition; also ζ′(1−s)/ζ(1−s)=O(1) because Re⁡(1−s)≥2.

givenalgebra
2.1

Stirling's formula bounds the logarithmic derivative of the Gamma factor by Oη(log⁡(∣s∣+2)) there. Combining the finitely many factor bounds proves the assertion.

step 1.1algebra∎

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