False statementConstruction: AI-adaptedVerification: AI-generatedverified 2026-09-26 (gpt-6-sol)

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: $ζ (- 1)$ 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 ordinary meaning of an infinite-series sum as the limit of its partial sums.

Refutation

technique · direct

1.1givenalgebra

The $N$ th partial sum of $1 + 2 + 3 + \dots$ is $N (N + 1) / 2$ , which tends to $+ \infty$ as $N \to \infty$ . Thus the series has no ordinary sum.

2.1step 1.1∎

Since the right-hand side has no ordinary sum, it cannot equal any value assigned to $ζ (- 1)$ by analytic continuation. This refutes the claim independently of that value.

Used by

Nothing in the library uses this result yet.

Dependency tree · 0 levels

Nothing. This result depends on no other item in the library.

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+⋯