CorollaryStatement: Literature-sourcedProof: AI-generatedPipeline-generatedprecheck passverified 2026-09-24 (gpt-6-sol)

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A unit-interval bound for zeta zeros

Statement

Assume countable choice.

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: Countable choice and the Riemann--von Mangoldt estimate.

1.1givenalgebra

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.

2.1step 1.1cases∎

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.

Depends on

Used by

Zeta zero count near the one line Corollary
Selecting an admissible contour height Example
A local formula for the logarithmic derivative of zeta Lemma
Zeta logarithmic derivative zero bound Lemma
Zeta reciprocal zero sum bound Lemma
A smoothed von Mangoldt explicit formula Theorem
The truncated von Mangoldt explicit formula Theorem

Dependency tree · two levels

15 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)