CorollaryStatement: Literature-sourcedProof: AI-generatedprecheck passaudited 2026-08-31

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Möbius inversion gives $Λ = μ * \log$

Statement

For every positive integer $n$ ,

$Λ (n) = \sum_{\begin{array}{c} d ∣ n \\ d > 0 \end{array}} μ (d) \log (n / d) .$

Equivalently,

$Λ = μ * \log .$

Facts & Assumptions

Given: A positive integer $n$ .

Proof

technique · direct

1.1given

By The divisor sum of von Mangoldt is the arithmetic-function logarithm, the arithmetic function $Λ$ satisfies $\log n = \sum_{d ∣ n} Λ (d)$ for every positive integer $n$ .

2.1step 1.1

Apply Classical Möbius inversion over positive divisors to the functions $f = Λ$ and $g = \log$ . It gives $Λ (n) = \sum_{d ∣ n} μ (d) \log (n / d)$ .

3.1step 2.1∎

The right-hand side is exactly the Dirichlet convolution formula for $μ * \log$ , so $Λ = μ * \log$ .

Depends on

Used by

The logarithmic derivative of the zeta Dirichlet series is the Dirichlet series of the von Mangoldt function on Re s greater than 1 Theorem

Dependency tree · two levels

11 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)

Möbius inversion gives Λ=μ∗log⁡Λ=μ∗log