An asymptotic formula for the number of totients

What OpenAI claims

Gives an asymptotic equivalent for the number V(x) of distinct totient values up to x, with a positive bounded phase-dependent factor determined by convergent arithmetic approximations. In particular, V(cx)/V(x) → c for every fixed c > 0, answering Erdős and Hall’s scaling question. (Lean)

Paper

1. An asymptotic formula for the number of totients

   25 September 2026 · Preprint, not peer reviewed · PDF

   Let V(x) count the distinct values of Euler's totient function up to x. We give an explicit asymptotic equivalent for V(x). Its coefficient is a uniform limit of functions defined from finite arithmetic data. We also prove that V(cx)/V(x) → c as x → ∞ for every fixed c > 0, answering a question of Erdős and Hall.

Status

Preprint. The paper has not been independently reviewed. Treat the result as OpenAI's claim until other mathematicians have checked it. If a video exists, it illustrates the statement; it is not the proof.