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

1.1

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 $\sum_{n \geq 1} Λ (n) n^{- σ} [3 + 4 \cos (t \log n) + \cos (2 t \log n)]$ .

2.1

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.

step 1.1algebra∎

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)