False statementConstruction: AI-adaptedVerification: 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.

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FALSE: zeta is given by the same Dirichlet series for every complex $s$ other than $1$

Statement

False claim: zeta is given by the Dirichlet series $\sum_{n \geq 1} n^{- s}$ for every complex $s \neq 1$ .

Facts & Assumptions

Given: The two concrete failures of that claim.

[L1]

At $s = 1 / 2$ , the eta series represents the continued zeta value while the Dirichlet series diverges (The eta series can represent the continued zeta function where the defining Dirichlet series diverges).

[L2]

At $s = 1$ , the defining Dirichlet series is the divergent harmonic series (The defining Dirichlet series for zeta diverges at $s = 1$ because it becomes the harmonic series).

Refutation

technique · direct

1.1

By [L1], the claim already fails at $s = 1 / 2$ : the continuation exists there, but the Dirichlet series does not converge.

L1given

2.1

By [L2], the excluded point $s = 1$ is also a divergence point for the defining series. Therefore the slogan "the same Dirichlet series works for every $s \neq 1$ " is false.

step 1.1L2algebra∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

5 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

Elias M. Stein and Rami Shakarchi, Complex Analysis, Ch. 6 §2.1 (standard reference, not scraped)

FALSE: zeta is given by the same Dirichlet series for every complex ss other than 11