Alphabeta Math
CorollaryStatement: Literature-sourcedProof: AI-generatedprecheck passaudited 2026-09-04
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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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The divisor-counting Dirichlet series is the square of the zeta Dirichlet series on Re s greater than 1

Statement

For ℜs>1,

∑n≥1τ(n)n−s=ζ(s)2,

where ζ(s)=∑n≥1n−s.

Facts & Assumptions

Given: A complex number s with ℜs>1.

[L1]

The divisor-counting function satisfies τ=1∗1 (The divisor functions arise by Dirichlet convolution, The divisor-counting function τ).

[L2]

Products of absolutely convergent Dirichlet series multiply by Dirichlet convolution (Multiplying absolutely convergent Dirichlet series gives Dirichlet convolution).

Proof

technique · direct
1.1L2givenalgebra

Applying [L2] to the constant-one function gives ζ(s)2=∑n≥1(1∗1)(n)n−s.

2.1L1step 1.1algebra∎

By [L1], the coefficient (1∗1)(n) is exactly τ(n), so step 1.1 is the claimed identity.

Depends on

Used by

Dependency tree · two levels

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Sources