| # An asymptotic formula for the number of totients | |
| The following describes the scope of the Lean formalization related to the following accompanying paper(s): | |
| - [An asymptotic formula for the number of totients](../../preprints/An-asymptotic-formula-for-the-number-of-totients-September-25-2026/An-asymptotic-formula-for-the-number-of-totients-September-25-2026.pdf) | |
| ## 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](../ComparatorChallenges/TotientAsymptotic.lean) | | |
| | Exact zero case for the companion totient counts | [TotientCompanionZero.lean](../ComparatorChallenges/TotientCompanionZero.lean) | | |