Alphabeta Math
CounterexampleConstruction: 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 Euler-product identity does not survive after leaving the absolute half-plane

Statement refuted

Once a multiplicative Dirichlet series is written as an Euler product on its absolute half-plane, the same prime-by-prime regrouping remains valid on the boundary or beyond.

Facts & Assumptions

Given: The Dirichlet series ζ(s)=∑n≥1n−s.

[L1]

The Euler-product theorem is stated only on a half-plane of absolute convergence (A multiplicative Dirichlet series factors as an Euler product on its absolute half-plane, Absolute convergence at one point forces absolute and locally uniform convergence on closed half-planes to the right).

Counterexample

technique · direct
1.1

On ℜs>1, [L1] gives ζ(s)=∏p(1−p−s)−1. At the boundary point s=1, however, the Dirichlet series is the harmonic series and diverges. So there is no value of the left-hand side there to which the proved Euler-product identity could apply.

L1givenalgebra
2.1

This already refutes the claimed boundary extension: the theorem proving the Euler product does not license prime-factor regrouping once absolute convergence is lost.

step 1.1∎

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