CorollaryStatement: Literature-sourcedProof: AI-generatedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-07

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A unit-interval bound for zeta zeros

Statement

The number of nontrivial zeta zeros, with multiplicity, whose ordinates lie in $[T, T + 1]$ is $O (\log (T + 2))$ for $T \geq 0$ .

Proof

Given: the Riemann--von Mangoldt estimate.

1.1

For $T \geq 3$ , the zeros with ordinates in $[T, T + 1]$ are included in $N (T + 2) - N (T - 1)$ . Subtract the Riemann--von Mangoldt formula at $T + 2$ and $T - 1$ ; its main term changes by $O (\log T)$ and its two errors have that size.

givenalgebra

2.1

On the compact range $0 \leq T < 3$ , discreteness of the zeros gives a fixed finite bound, which is absorbed by enlarging the constant. The buffered count in step 1.1 already includes both endpoints, so the stated closed interval is covered.

step 1.1cases∎

Depends on

The Riemann--von Mangoldt zero count

Used by

Selecting an admissible contour height Example
A local formula for the logarithmic derivative of zeta Lemma
A smoothed von Mangoldt explicit formula Theorem
The truncated von Mangoldt explicit formula Theorem

Dependency tree · two levels

9 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, Lemma 10.3 (standard reference, not scraped)