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 average order of Euler's totient is 6n over pi squared

Statement

The arithmetic function $n \mapsto 6 n / π^{2}$ is an average order of $φ$ .

Facts & Assumptions

Given: A real $x \geq 1$ and $N : = ⌊ x ⌋$ .

Proof

technique · direct

1.1

The comparison sum satisfies $\sum_{n \leq x} \frac{6 n}{π^{2}} = \frac{6}{π^{2}} \sum_{n = 1}^{N} n = \frac{3}{π^{2}} N (N + 1) = \frac{3}{π^{2}} x^{2} + O (x) .$

givenalgebra

2.1

By The summatory totient function is 3 over pi squared times x squared plus O(x log x), $\sum_{n \leq x} φ (n) = 3 x^{2} / π^{2} + O (x \log x)$ . Comparing with step 1.1 shows that the two summatory functions are asymptotic, with the comparison sum eventually positive. Hence Summatory functions and average orders says that $6 n / π^{2}$ is an average order of $φ$ .

step 1.1givenalgebra∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

10 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)