LemmaStatement: AI-adaptedProof: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-07

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Zeta three four one logarithmic derivative inequality

Statement

For $σ > 1$ and $t \in R$ , $- 3 \frac{ζ^{'} (σ)}{ζ (σ)} - 4 Re \frac{ζ^{'} (σ + i t)}{ζ (σ + i t)} - Re \frac{ζ^{'} (σ + 2 i t)}{ζ (σ + 2 i t)} \geq 0.$

Facts & Assumptions

Given: The data and hypotheses of the statement.

[F1]

The logarithmic derivative of the zeta Dirichlet series is the Dirichlet series of the von Mangoldt function on Re s greater than 1: For $ℜ s > 1$ , if $ζ (s) : = \sum_{n \geq 1} n^{- s},$ then $- \frac{ζ^{'} (s)}{ζ (s)} = \sum_{n \geq 1} Λ (n) n^{- s} .$

Proof

1.1F1

The negative logarithmic derivative has coefficients $Λ (n) \geq 0$ . Its series is absolutely convergent for $σ > 1$ (also $Λ (n) \leq \log n$ ), so the displayed expression equals $\sum_{n \geq 1} Λ (n) n^{- σ} [3 + 4 \cos (t \log n) + \cos (2 t \log n)]$ .

2.1step 1.1algebra∎

For real $u$ , $3 + 4 \cos u + \cos (2 u) = 2 (1 + \cos u)^{2} \geq 0$ . Every summand is nonnegative, so the convergent sum is nonnegative.

Depends on

The logarithmic derivative of the zeta Dirichlet series is the Dirichlet series of the von Mangoldt function on Re s greater than 1

Used by

The three four one trigonometric inequality Example
Riemann zeta classical zero free region Theorem

Dependency tree · two levels

5 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

§6.1, proof of Theorem 6.6 invoking Lemma 6.5 (standard reference, not scraped)