OpenAI claim summary for Math-2779

An explicit asymptotic formula for the number of totients

OpenAI's claim

OpenAI presents the following claims relevant to this question, with manuscript numbers following paper order in Sources. In manuscript 1 (family 024; 2026-09-25), the asserted result is that distinct totient values admit an explicit asymptotic with a coefficient obtained from finite arithmetic approximations, and the count is regularly varying with index one.

Mathematical statement of the claimed result

Manuscript 1.

Let $V(x)=|\{\varphi(k):k\ge1,\ \varphi(k)\le x\}|$, counting distinct Euler-totient values. All logarithms below are natural. Put \[ a_j=(j+1)\log(j+1)-j\log j-1,\quad \sum_{j\ge1}a_j\rho^j=1\ (0<\rho<1),\quad\lambda=\log(1/\rho), \quad g_0=1,\quad g_j=\sum_{d=1}^ja_dg_{j-d}. \] For sufficiently large $x$, define $B=\log\log x$, $u=(\log B-\log\log B)/\lambda$, $m=\lfloor u\rfloor$, $\theta=u-m$, and $G_m=B^m/(m!\prod_{j=1}^mg_j)$. The claim constructs a phase function $A:[0,1)\to(0,\infty)$ as a uniform limit of finite arithmetic coefficients, with $0<\inf A\le\sup A<\infty$, and asserts \[ V(x)\sim\frac{xG_m}{\log x}A(\theta),\qquad \forall c>0:\quad\lim_{x\to\infty}\frac{V(cx)}{V(x)}=c. \] No continuity of $A$ is asserted. The subscripted logarithms in the manuscript mean iterated logarithms, not logarithms to an integer base.

Outline of the argument

For manuscript 1 (family 024; 2026-09-25), long prime prefixes and smooth cofactors cover almost all values. A collision estimate identifies when different shortened tuples represent the same totient, and a sieve comparison determines the main term after those collisions are removed.

Formal verification

For manuscript 1 (family 024; 2026-09-25), OpenAI lists Lean statements for this assertion: distinct totient values admit an explicit asymptotic with a coefficient obtained from finite arithmetic approximations, and the count is regularly varying with index one.

The central assertion is within the documented formal scope; the additional applications retain their own scope limits.

The linked family Comparator checks are TotientAsymptotic.lean, TotientCompanionZero.lean.

Their named declarations include OAI.TotientAsymptotic.coefficient_nonnegative, OAI.TotientAsymptotic.companion_zero_case, OAI.TotientAsymptotic.totient_asymptotic_formula, OAI.TotientAsymptotic.weighted_totient_asymptotic, OAI.TotientAsymptotic.weighted_totient_one_two.

This site has not executed the Lean code. Described statement coverage is not an independent proof check.

Status caveat

The manuscripts were produced by an internal OpenAI model. OpenAI cautions that results without formalization can contain mistakes. This site has not independently checked the proofs; the label records the scope of OpenAI's claim.

Sources

• OpenAI, An asymptotic formula for the number of totients, 2026-09-25; citation key OAI:An-asymptotic-formula-for-the-number-of-totients-September-25-2026. PDF (GitHub); PDF (direct); Manuscript page and citation; Latest version; Statement source: thm:main.
• Family 024: Lean scope.
• Comparator TotientAsymptotic.json.
• Comparator TotientAsymptotic.lean.
• Comparator TotientCompanionZero.json.
• Comparator TotientCompanionZero.lean.
• OpenAI catalogue entry.