Alphabeta Math
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.

The Dirichlet series of the Liouville function is zeta of 2s divided by zeta of s

Example

For ℜs>1,

∑n≥1λ(n)n−s=ζ(2s)ζ(s),

where λ is the Liouville function.

Facts & Assumptions

Given: A complex number s with ℜs>1.

[L1]

The Liouville function satisfies λ(pk)=(−1)k (Liouville's function).

[L2]

Completely multiplicative Dirichlet series have geometric Euler factors (Completely multiplicative Dirichlet series have geometric Euler factors).

Verification

technique · direct
1.1

By [L2], ∑n≥1λ(n)n−s=∏p11+  p−s. Indeed [L1] gives the local ratio f(p)=−1, so the denominator is 1−(−p−s)=1+p−s.

L1L2givenalgebra
2.1

Since 11+p−s=1−p−s1−p−2s, multiplying over primes gives ∏p1−p−s1−p−2s=ζ(2s)ζ(s). So the claimed identity holds.

step 1.1algebra∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

6 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