ExampleConstruction: Literature-sourcedVerification: AI-generatedprecheck passaudited 2026-09-04

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.

A local prime factor in the Dirichlet series of Euler's totient

Example

For a prime $p$ , the $p$ -power contribution to the Dirichlet series of $φ$ is

$\sum_{k \geq 0} φ (p^{k}) p^{- k s} = \frac{1 - p^{- s}}{1 - p^{1 - s}} .$

Facts & Assumptions

Given: A prime $p$ and $ℜ s > 2$ .

[L1]

The Dirichlet series of $φ$ is $ζ (s - 1) / ζ (s)$ (The Dirichlet series of Euler's totient is zeta of s minus 1 divided by zeta of s on Re s greater than 2).

[L2]

$φ (p^{k}) = p^{k} - p^{k - 1}$ for $k \geq 1$ (For a prime $p$ and $k \geq 1$ , $φ (p^{k}) = p^{k} - p^{k - 1}$ ).

Verification

technique · direct

1.1L2givenalgebra

By [L2], $\sum_{k \geq 0} φ (p^{k}) p^{- k s} = 1 + \sum_{k \geq 1} (p^{k} - p^{k - 1}) p^{- k s} = 1 + (1 - p^{- 1}) \sum_{k \geq 1} p^{k (1 - s)} .$

2.1L1step 1.1algebra∎

Summing the geometric series gives $1 + (1 - p^{- 1}) \frac{p^{1 - s}}{1 - p^{1 - s}} = \frac{1 - p^{- s}}{1 - p^{1 - s}} .$ This is the local factor of [L1].

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

19 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, Chapter 2 examples (standard reference, not scraped)