DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedverified 2026-09-24 (gpt-6-sol)

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The Riemann zeta zero-counting function

Definition

Assume countable choice.

For $T > 0$ , $N (T)$ is the number, with multiplicity, of nontrivial zeros $ρ = β + i γ$ of the meromorphic continuation of zeta satisfying $0 < γ \leq T$ . Thus a zero on the top boundary is included.

Depends on

Used by

A local formula for the logarithmic derivative of zeta Lemma
Zeta reciprocal zero sum bound Lemma
The Riemann--von Mangoldt zero count Theorem

Dependency tree · two levels

13 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

Nick Andersen, Analytic Number Theory, §11.2 (standard reference, not scraped)