An asymptotic formula for the number of totients
The following describes the scope of the Lean formalization related to the following accompanying paper(s):
Scope
Let $V(x)$ count the distinct values of Euler's totient function up to $x$. The formalization constructs the paper's explicit positive main term from finite arithmetic approximants and proves that their ratio tends to one. In particular, $V(cx)/V(x)\to c$ for every fixed $c>0$, answering the Erdős–Hall regular-variation question.
The formalization also gives asymptotics for totients $v\le x$ whose least preimage $\ell(v)=\min{n\ge1:\varphi(n)=v}$ lies between $kx$ and $(k+1)x$. The associated coefficient is positive under the stated existence condition; when no totient $d$ satisfies $kd<\ell(d)$, the count and coefficient are identically zero. The cases $k=1,2$ have positive coefficients.
Comparator links
| Result | Comparator statement |
|---|---|
| Totient-count asymptotics and regular variation | TotientAsymptotic.lean |
| Exact zero case for the companion totient counts | TotientCompanionZero.lean |