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How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Checking the von Mangoldt divisor-sum identity on sample integers

Example

For $16$ and $30$ the divisor-sum identity reads

$\sum_{d ∣ 16} Λ (d) = 4 \log 2 = \log 16$

and

$\sum_{d ∣ 30} Λ (d) = \log 2 + \log 3 + \log 5 = \log 30.$

Facts & Assumptions

Given: The integers $16$ and $30$ .

Verification

technique · direct

1.1givenalgebra

The divisors of $16$ are $1, 2, 4, 8, 16$ . By The von Mangoldt function, only the prime powers $2, 4, 8, 16$ contribute, each with value $\log 2$ , so $\sum_{d ∣ 16} Λ (d) = 4 \log 2 = \log 16$ .

1.2givenalgebra

The divisors of $30$ are $1, 2, 3, 5, 6, 10, 15, 30$ . Again by The von Mangoldt function, only $2, 3, 5$ contribute, so $\sum_{d ∣ 30} Λ (d) = \log 2 + \log 3 + \log 5 = \log 30$ .

2.1step 1.1step 1.2∎

These are exactly the values predicted by The divisor sum of von Mangoldt is the arithmetic-function logarithm on one prime power and one squarefree composite with three prime factors.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

8 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)