Alphabeta Math
CorollaryStatement: Literature-sourcedProof: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-31
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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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The average order of tau is log n

Statement

The arithmetic function n↦log⁡n is an average order of τ.

Facts & Assumptions

Given: A real x≥1.

Proof

technique · direct
1.1

By The summatory divisor-counting function is x log x plus (2 gamma - 1)x plus O(sqrt x) and The summatory logarithm is x log x minus x plus O(log x), ∑n≤xτ(n)=xlog⁡x+(2γ−1)x+O(x),∑n≤xlog⁡n=xlog⁡x−x+O(log⁡x).

given
2.1

The two sums differ by 2γx+O(x), while the comparison sum is xlog⁡x+O(x) and is therefore eventually positive. Hence ∑n≤xτ(n)∑n≤xlog⁡n=1+O ⁣(1log⁡x)→1. By Summatory functions and average orders, this is exactly the statement that log⁡n is an average order of τ.

step 1.1givenalgebra∎

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