CorollaryStatement: Literature-sourcedProof: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-31

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
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The proportion of pairs in {1,...,n}^2 that are coprime tends to 6 over pi squared

Statement

For positive integers $n$ , the proportion of pairs in ${1, \dots, n}^{2}$ that are coprime tends to $6 / π^{2}$ .

Facts & Assumptions

Given: A positive integer $n$ .

Proof

technique · direct

1.1

By Ordered coprime pairs in a box have asymptotic density 6 over pi squared, the number of coprime pairs in ${1, \dots, n}^{2}$ is $\frac{6}{π^{2}} n^{2} + O (n \log n) .$

given

2.1

Dividing by the total number $n^{2}$ of ordered pairs gives $\frac{# {(a, b) \in {1, \dots, n}^{2} : \gcd (a, b) = 1}}{n^{2}} = \frac{6}{π^{2}} + O ⁣ (\frac{\log n}{n}),$ and the error term tends to $0$ .

step 1.1givenalgebra∎

Depends on

Ordered coprime pairs in a box have asymptotic density 6 over pi squared

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

3 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

Karl-Dieter Crisman, Number Theory: In Context and Interactive (standard reference, not scraped)