OpenAI · 024-01 · 2026-09-25 · Number theory
An asymptotic formula for the number of totients
preprints/An-asymptotic-formula-for-the-number-of-totients-September-25-2026
Family 024. An asymptotic formula for the number of totients.
The authors are OpenAI. This companion does not verify the proof and does not adopt it as a theorem of Ngo Cheung’s series. It is not a premise. O.1 stays NOT_CLOSED. Anti-de Sitter results are not a de Sitter state, not Em02-E’s relational lapse, not XVI-P’s QdS scheme, and not XVI-Q’s sigma-image record. No quantum coefficient is admitted. No compilation is asserted.
Read against the standing problems
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
Gives an asymptotic equivalent for the number of distinct totient values up to , with a positive bounded phase-dependent factor determined by convergent arithmetic approximations. In particular, for every fixed , answering Erdős and Hall’s scaling question.
Let count the distinct values of Euler's totient function up to x. We give an explicit asymptotic equivalent for . Its coefficient is a uniform limit of functions defined from finite arithmetic data. We also prove that as for every fixed c > 0, answering a question of Erdős and Hall.
Preprint source and Lean scope
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This preprint is named in the Lean scope note for 024: An asymptotic formula for the number of totients. The authors are OpenAI. This companion does not verify the formalization. It is not a premise. The OAI library is not imported. No compilation is asserted. O.1 stays NOT_CLOSED.