An asymptotic formula for the number of totients

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)\to c for every fixed c > 0, answering Erdős and Hall’s scaling question. (Lean)

Manuscript

An asymptotic formula for the number of totients

September 25, 2026 · source · cite as \cite{OAI:An-asymptotic-formula-for-the-number-of-totients-September-25-2026}

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)\to c as x\to\infty for every fixed c > 0, answering a question of Erdős and Hall.

Result 024 · Number theory. Imported from openai/math, a collection of manuscripts produced by an OpenAI model; results are at different stages of verification. Discussion below is by Lemma members.