Alphabeta Math
LemmaStatement: AI-adaptedProof: AI-adaptedPipeline-generatedprecheck passverified 2026-09-24 (gpt-6-sol)
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Zeta horizontal logarithmic derivative comparison

Statement

Assume countable choice.

There are absolute d>0,C>0, with d<c0, such that for ∣t∣≥3 and σ≥1−d/log⁡(∣t∣+2), ∣ζ′(σ+it)ζ(σ+it)∣≤Clog⁡(∣t∣+2).

Facts & Assumptions

Given: Countable choice and the data and hypotheses of the statement.

[F1]

Riemann zeta classical zero free region: There is an absolute c0>0 such that ζ has no zeros in σ≥1−c0/log⁡(∣t∣+2). The pole at s=1 is not a zero.

[F2]

A local formula for the logarithmic derivative of zeta: Uniformly for −1≤σ≤2 and s=σ+it≠1 whose ordinate is not that of a nontrivial zero, ζ′ζ(s)=−1s−1+∑ρ: ∣t−ℑρ∣<11s−ρ+O(log⁡(∣t∣+2)), where zeros occur with multiplicity. For ∣t∣≥2 the pole term is absorbed into the error, giving the usual large-height local formula.

[F3]

Zeta logarithmic derivative zero bound: Write s=σ+it and let ρ range over nontrivial zeta zeros with multiplicity. With the Hadamard constant B, ζ′ζ(s)=B+∑ρ(1s−ρ+1ρ)−1s−1+log⁡π2−Γ′(1+s/2)2Γ(1+s/2). This is a meromorphic identity, using convergent genus-one terms. Uniformly for −1≤σ≤2, ∣t∣≥3 and ζ(s)≠0, Re⁡ζ′ζ(s)=∑ρRe⁡1s−ρ−12log⁡∣t∣+O(1), and this real series is absolutely convergent.

[F4]

The logarithmic derivative of the zeta Dirichlet series is the Dirichlet series of the von Mangoldt function on Re s greater than 1: For ℜs>1, if ζ(s):=∑n≥1n−s, then −ζ′(s)ζ(s)=∑n≥1Λ(n)n−s.

Proof

1.1F3F4

Under countable choice, the supplied zero-bound identity and zero-free region apply. Put L=log⁡(∣t∣+2), s1=1+L−1+it. The Euler series and the real-axis simple-pole expansion give ∣ζ′/ζ(s1)∣≤−ζ′/ζ(1+L−1)=O(L). The same comparison holds for every σ≥1+L−1; for σ≥2 it is even bounded by the convergent series at two. The real-part formula now gives ∑ρRe⁡(1/(s1−ρ))=O(L), all summands being positive.

1.2F1algebra

For ∣Im⁡ρ−t∣≤1, the region theorem implies 1−Re⁡ρ≥c0/(KL) with an absolute K, since log⁡(∣Im⁡ρ∣+2)≤KL. Choose d<c0/(2K). For 1−d/L≤σ≤1+1/L, the positive real parts of s−ρ and s1−ρ are comparable, hence ∣s−ρ∣≥c∣s1−ρ∣. Consequently ∣1/(s−ρ)−1/(s1−ρ)∣≤C/(L∣s1−ρ∣2)≤CRe⁡(1/(s1−ρ)).

2.1F2step 1.1step 1.2∎

For ordinates off the zero ordinates, subtract the two local logarithmic-derivative formulas. Their pole terms are bounded at these heights. Sum the preceding comparison over the common local zero set and use its positive-sum bound to obtain O(L). The constants do not depend on the distance of t from an ordinate. Taking limits from non-ordinates extends the bound to all t, because the entire horizontal segment is zero-free. Together with the Euler-series range this proves the assertion.

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