Alphabeta Math
ExampleConstruction: Literature-sourcedVerification: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-04
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The first coefficients of 1 over zeta are the Möbius values

Example

The reciprocal-zeta identity begins

1ζ(s)=1−12s−13s−15s+16s−17s+O(8−s),

matching the first Möbius values.

Facts & Assumptions

Given: The reciprocal-zeta identity on ℜs>1.

[L1]

The Möbius Dirichlet series is 1/ζ(s) (The Dirichlet series of the Möbius function is the reciprocal of the zeta Dirichlet series on Re s greater than 1).

[L2]

Dirichlet-series multiplication is convolution (Multiplying absolutely convergent Dirichlet series gives Dirichlet convolution).

Verification

technique · direct
1.1

The first Möbius values are μ(1)=1, μ(2)=μ(3)=μ(5)=μ(7)=−1, μ(4)=μ(8)=0, μ(6)=1. So the displayed initial segment is exactly ∑n≤7μ(n)n−s.

L1givenalgebra
2.1

Multiplying this initial segment by the initial segment of ζ(s) and using [L2], the coefficients through n=7 cancel to those of the Dirichlet-convolution identity. That is the coefficient-level content of [L1].

L1L2step 1.1algebra∎

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