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

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The divisor sum of von Mangoldt is the arithmetic-function logarithm

Statement

For every positive integer $n$ ,

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

Equivalently, if the arithmetic function $\log$ is defined by $n \mapsto \log n$ , then

$1 * Λ = \log .$

Facts & Assumptions

Given: A positive integer $n$ .

Proof

technique · direct

1.1givenalgebra

Write the canonical prime factorization of $n$ as $n = \prod_{i < r} p_{i}^{e_{i}}$ using For $n \geq 1$ and any injective list $p : r \to Z$ of primes containing every prime divisor of $n$ , one has $n = \prod_{i < r} p_{i}^{v_{p_{i}} (n)}$ ; the exponents are determined by $n$ , and $v_{q} (n) = 0$ for every prime $q$ outside the list. A positive divisor of $n$ contributes to the sum only when it is a prime power $p_{i}^{j}$ , because The von Mangoldt function is zero on every other divisor. Thus $\sum_{d ∣ n} Λ (d) = \sum_{i < r} \sum_{j = 1}^{e_{i}} \log p_{i} = \sum_{i < r} e_{i} \log p_{i}$ .

2.1step 1.1algebra

Repeatedly applying the product law from Order, continuity, range, and the product, quotient, and reciprocal laws for the natural logarithm gives $\log n = \log (\prod_{i < r} p_{i}^{e_{i}}) = \sum_{i < r} \log (p_{i}^{e_{i}}) = \sum_{i < r} e_{i} \log p_{i}$ . For $n = 1$ both sums are empty, so the same formula gives $0 = \log 1$ .

3.1step 1.1step 2.1∎

Comparing the two expressions in steps 1.1 and 2.1 proves the divisor-sum identity, and the convolution form is exactly Dirichlet convolution of arithmetic functions with the constant-one function.

Depends on

Used by

Möbius inversion gives Λ=μ*log Corollary
Checking the von Mangoldt divisor-sum identity on sample integers Example
The von Mangoldt harmonic sum is log x plus O(1) Lemma
Mertens sum for primes in an arithmetic progression Theorem

Dependency tree · two levels

40 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

Victor Shoup, A Computational Introduction to Number Theory and Algebra, Exercise 2.51 (standard reference, not scraped)
Tom Sanders, Topics in Analytic Number Theory, Chapter 1 (standard reference, not scraped)