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The normal order of the number of iterations of Euler’s function

Plasma AI

Let f(n) count the iterations of Euler’s totient function needed to reach 1. The manuscript reports a constant α between 1/log 3 and 1/log 2 such that f(n)/log n tends to α on a set of natural density one. It also gives a quantitative mean-square bound, answering the first two parts of Erdős Problem 408.

Sources for this record

What was already known

Erdős, Granville, Pomerance and Spiro obtained the normal order under Elliott–Halberstam-type hypotheses. This argument uses their moment method with new quantitative shifted-prime estimates derived from the announced zero-free-half-plane and weighted dilation-graph results.

AI involvement AI-led

The manuscript credits Plasma AI as institutional author. The project records Claude Opus 5.5 agents extending the graph inputs, simplifying the rectangle-statistic deduction and constructing the formal proofs under human direction. This supports AI-led work; task-only autonomy for the original argument is not established.

Importance Notable

Removes the earlier conditional distribution assumptions for the specified normal-order questions by deriving the needed quantitative estimates from the stated upstream results.

Limitations

The value of α is not determined, and the broader largest-prime-factor question in Problem 408 is not resolved. The proof depends on the announced OpenAI zero-free-half-plane and graph estimates and their vendored formalizations. The manuscript uses f(1)=0; the compared problem convention uses f(1)=1, which does not affect the normal order.

Technical verification Formal (Lean)

Palomar reports comparison of the three selected statements and Lean, NanoDa and con-ron kernel acceptance on 19 October 2026. This is a source-reported check of the pinned formal statement.

Formal checking

Palomar reports comparison of the three selected statements and Lean, NanoDa and con-ron kernel acceptance on 19 October 2026. This is a source-reported check of the pinned formal statement.

Checker
Palomar: Lean, NanoDa and con-ron
Scope
The quantitative normal-order theorem and the first two formal-conjectures Erdős 408 statements, with only propext, Classical.choice and Quot.sound; includes the vendored upstream developments.
Basis
Reported by the linked checker
Evidence source

Checking a formal statement does not establish that it matches the original problem.

Human understanding Few

No named human account explaining the scoped central reasoning was established. Few is the documentation default, not a finding that people cannot understand it.

Correction and publication history

2026-10-20 · publication

Added the sourced result with its exact hypotheses, prior work, AI contribution and checking scope.

Read this published revision