DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-5.6-terra)audited 2026-08-31

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The von Mangoldt function

Definition

The von Mangoldt function is the arithmetic function $Λ$ defined by

Λ (n) : = {\begin{cases} \log p, & n = p^{k} for some prime p and some k \geq 1, \\ 0, & otherwise. \end{cases}

Remarks

The prime $p$ in the first case is unique by The fundamental theorem of arithmetic: every integer $n \geq 1$ is a product of primes, and the factorisation is unique up to order — if $\prod_{i < r} p_{i} = \prod_{j < s} q_{j}$ with every $p_{i}$ and $q_{j}$ prime, then $r = s$ and $q_{i} = p_{π (i)}$ for some $π \in Sym (r)$ , so the definition does not depend on a choice.

Depends on

Used by

Dependency tree · two levels

30 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, An Introduction to Analytic Number Theory, Definition 3.9 (standard reference, not scraped)
Tom Sanders, Topics in Analytic Number Theory, Chapter 1 (standard reference, not scraped)