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.

A short Euler-product truncation already numerically approximates zeta at $2$

Example

Using the first four primes,

$\prod_{p \in {2, 3, 5, 7}} \frac{1}{1 - p^{- 2}} = \frac{1225}{768} \approx 1.59505,$

while

$ζ (2) = \frac{π^{2}}{6} \approx 1.64493.$

Facts & Assumptions

Given: The Euler product and the independent Basel sum.

[L1]

For $Re s > 1$ , $ζ (s) = \prod_{p} (1 - p^{- s})^{- 1}$ (The Riemann zeta function has its Euler product on the half-plane $Re s > 1$ ).

[L2]

The Basel sum is $\sum_{n \geq 1} n^{- 2} = π^{2} / 6$ (The Basel sum is pi squared over six by a residue computation), and this convergent Dirichlet series is $ζ (2)$ (The Riemann zeta function on the half-plane $Re s > 1$ ).

Verification

technique · direct

1.1L1givenalgebra

Evaluating the Euler factors from [L1] at $s = 2$ gives ${(1 - \frac{1}{4})}^{- 1} {(1 - \frac{1}{9})}^{- 1} {(1 - \frac{1}{25})}^{- 1} {(1 - \frac{1}{49})}^{- 1} = \frac{4}{3} \cdot \frac{9}{8} \cdot \frac{25}{24} \cdot \frac{49}{48} = \frac{1225}{768} .$

2.1step 1.1L2algebra∎

By [L2], the target value is $π^{2} / 6 \approx 1.64493$ . Comparing with step 1.1 shows that even this short prime truncation already lands within about five hundredths of $ζ (2)$ .

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

13 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 (standard reference, not scraped)

A short Euler-product truncation already numerically approximates zeta at 22