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Math-2779 · mathematics · Number theory

An explicit asymptotic formula for the number of totients

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
Determine a fully specified asymptotic equivalent for \[ V(x)=\#\{m\leq x:m=\phi(n)\text{ for some positive integer }n\}, \] counting each distinct totient value once.

Short English statement

Determine a fully specified asymptotic equivalent for the number $V(x)=\#\{m\leq x:m=\phi(n)\text{ for some positive integer }n\}$ of distinct totient values at most $x$, counting each value once.

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2779-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2779

The result supplies the conclusion in this problem with its displayed hypotheses and quantifiers. The broader statements and applications in the manuscript are kept separate from the particular question above. The supporting result is manuscript 1 (family 024; 2026-09-25).

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

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