Alphabeta Math
CounterexampleConstruction: AI-generatedVerification: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-31
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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.

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Tau is multiplicative but not completely multiplicative

Statement refuted

The statement "every multiplicative arithmetic function is completely multiplicative" is false.

Facts & Assumptions

Given: The divisor-counting function τ.

Counterexample

technique · direct
1.1

By The divisor functions arise by Dirichlet convolution, the function τ is multiplicative in the sense of Multiplicative arithmetic functions.

given
2.1

The same proposition gives τ(2)=2 and τ(4)=3. Hence τ(4)=3≠4=τ(2)2, so τ(mn)=τ(m)τ(n) fails at the non-coprime pair m=n=2.

step 1.1algebra
3.1

Therefore τ is a multiplicative arithmetic function that is not completely multiplicative, contradicting the refuted statement.

step 2.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

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Sources