False statementConstruction: AI-adaptedVerification: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 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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FALSE: $ζ (- 1)$ is the ordinary sum $1 + 2 + 3 + \dots$

Statement

False claim: $ζ (- 1)$ is the ordinary sum $1 + 2 + 3 + \dots$ .

Facts & Assumptions

Given: The special value at $- 1$ .

[L1]

The special-values theorem gives $ζ (- 1) = - 1 / 12$ (The Riemann zeta function has the standard Bernoulli special values at the positive even and nonpositive integers).

[L2]

The continuation remark records that this value is not an ordinary series sum (The analytic continuation of zeta is not the same object as the defining Dirichlet series outside $Re s > 1$ ).

Refutation

technique · direct

1.1

By [L1], the continued zeta value at $- 1$ is $- 1 / 12$ .

L1given

2.1

The ordinary partial sums of $1 + 2 + 3 + \dots$ are $1, 3, 6, \dots$ , so they do not equal the fixed number $- 1 / 12$ . Step 1.1 and [L2] therefore refute the claim.

step 1.1L2algebra∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

7 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

K. Chandrasekharan, Lectures on the Riemann Zeta-Function, Lecture 11 §3 (standard reference, not scraped)

FALSE: ζ(−1)ζ(−1) is the ordinary sum 1+2+3+⋯1+2+3+⋯