ExampleConstruction: AI-generatedVerification: AI-generatedprecheck passverified 2026-09-24 (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.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

The special-value formula gives $ζ (4) = π^{4} / 90$

Example

$ζ (4) = \frac{π^{4}}{90} .$

Facts & Assumptions

Given: The Bernoulli generating function and the cotangent expansion.

[L1]

Bernoulli numbers are defined by $\frac{t}{e^{t} - 1} = \sum_{n \geq 0} \frac{B_{n}}{n!} t^{n}$ (The Bernoulli numbers are defined by the generating series $t / (e^{t} - 1)$ ).

[L2]

The cotangent expansion is $π \cot (π z) = 1 / z + \sum_{n \geq 1} 2 z / (z^{2} - n^{2})$ (The Mittag-Leffler expansion of pi cotangent).

[L3]

For $Re s > 1$ , $ζ (s) = \sum_{n \geq 1} n^{- s}$ (The Riemann zeta function on the half-plane $Re s > 1$ ).

Verification

technique · direct

1.1L1givenalgebra

Expanding $e^{t} - 1 = t + t^{2} / 2 + t^{3} / 6 + t^{4} / 24 + \dots$ and solving for the quotient in [L1] gives $\frac{t}{e^{t} - 1} = 1 - \frac{t}{2} + \frac{t^{2}}{12} - \frac{t^{4}}{720} + \dots,$ so $B_{4} = - 1 / 30$ .

2.1step 1.1L1L2L3algebra∎

For $∣ z ∣ < 1$ , expand each term of [L2] into an absolutely locally convergent geometric series and use [L3]. The coefficient of $z^{4}$ in $π z \cot (π z)$ is then $- 2 \sum_{n \geq 1} n^{- 4} = - 2 ζ (4)$ . On the other hand, setting $t = 2 π i z$ in [L1] gives $π z \cot (π z) = t / (e^{t} - 1) + t / 2$ ; the coefficient of $z^{4}$ is $B_{4} (2 π i)^{4} / 4! = - π^{4} / 45$ by step 1.1. Equating these coefficients yields $ζ (4) = π^{4} / 90$ .

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

12 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)

The special-value formula gives ζ(4)=π4/90ζ(4)=π4/90