Normal order of the number of Euler-totient iterations
Follow the latest progress by email.
Let f(n) be the number of iterations of Euler’s totient function required to reach 1. Does f(n)/log n tend to one constant on a set of natural density one, and what quantitative mean-square bound holds?
Why it matters and what counts as progress
Why it matters: A normal order gives the typical iteration count rather than only worst-case estimates.
What counts as progress: A density-one limit and quantitative moment control with all upstream shifted-prime distribution inputs justified; the constant’s interval alone does not prove the limit.
How the problem developed
- 1990 · Conditional result [1]
Paul Erdős, Andrew Granville, Carl Pomerance and Claudia A. Spiro Established the normal-order behavior of iterated Euler totients under an Elliott–Halberstam-type hypothesis.
- 14 Oct 2026 · Unconditional result [2]
Plasma AI Reported the unconditional normal order f(n) ∼ α log n for a constant α in [1/log 3, 1/log 2], with a quantitative variance bound and density-one conclusion.
AI contribution
The normal order of the number of iterations of Euler’s function
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
- Erdős, Granville, Pomerance and Spiro, On the normal behavior of the iterates of some arithmetic functions
- Plasma AI: normal order of totient iterations
- AI roles, upstream inputs and statement correspondence
- Exact normal-order and density statements
- Palomar: versioned checking record
- Erdős Problem 408: questions and prior work
Website timeline
Publication dates and editorial changes.
The normal order of the number of iterations of Euler’s function