ExampleConstruction: Literature-sourcedVerification: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-04

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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.
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The coefficient of 12 in zeta squared is the divisor count of 12

Example

The coefficient of $12^{- s}$ in $ζ (s)^{2}$ is $τ (12) = 6$ .

Facts & Assumptions

Given: The identity $\sum_{n \geq 1} τ (n) n^{- s} = ζ (s)^{2}$ .

[L1]

The square of the zeta Dirichlet series is the divisor-counting Dirichlet series (The divisor-counting Dirichlet series is the square of the zeta Dirichlet series on Re s greater than 1).

Verification

technique · direct

1.1

The positive divisors of $12$ are $1, 2, 3, 4, 6, 12$ , so $τ (12) = 6$ .

L1givenalgebra

2.1

Therefore [L1] says that the coefficient of $12^{- s}$ in $ζ (s)^{2}$ is $6$ .

L1step 1.1∎

Depends on

The divisor-counting Dirichlet series is the square of the zeta Dirichlet series on Re s greater than 1

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

4 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, Notes on Analytic Number Theory, Definition 2.8 (standard reference, not scraped)